Triangles - ACT Math

Card 0 of 1620

Question

Two similiar triangles exist where the ratio of perimeters is 4:5 for the smaller to the larger triangle. If the larger triangle has sides of 6, 7, and 12 inches, what is the perimeter, in inches, of the smaller triangle?

Answer

The larger triangle has a perimeter of 25 inches. Therefore, using a 4:5 ratio, the smaller triangle's perimeter will be 20 inches.

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Question

Two similiar triangles have a ratio of perimeters of $7:2$ .

If the smaller triangle has sides of 3, 7, and 5, what is the perimeter of the larger triangle.

Answer

Adding the sides gives a perimeter of 15 for the smaller triangle. Multipying by the given ratio of $\frac{7}{2}$ , yields 52.5.

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Question

Two similar triangles' perimeters are in a ratio of . If the lengths of the larger triangle's sides are , , and , what is the perimeter of the smaller triangle?

Answer

1. Find the perimeter of the larger triangle:

2. Use the given ratio to find the perimeter of the smaller triangle:

$\frac{2}{5}=\frac{x}{56}$

Cross multiply and solve:

$\frac{112}{5}=x=22.4$

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Question

There are two similar triangles. Their perimeters are in a ratio of . If the perimeter of the smaller triangle is , what is the perimeter of the larger triangle?

Answer

Use proportions to solve for the perimeter of the larger triangle:

$\frac{7}{9}=\frac{16.8}{x}$

Cross multiply and solve:

$\frac{151.2}{7}=x=21.6$

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Question

Two similar triangles have perimeteres in the ratio . The sides of the smaller triangle measure , , and respectively. What is the perimeter, in meters, of the larger triangle?

Answer

Since the perimeter of the smaller triangle is , and since the larger triangle has a perimeter in the ratio, we can set up the following identity, where the perimeter of the larger triangle:

$\frac{12}{x} = \frac{4}5{}$

In cross multiplying this identity, we get . We can now solve for . Here, , so the perimeter of the larger triangle is .

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Question

A circle contains 6 copies of a triangle; each joined to the others at the center of the circle, as well as joined to another triangle on the circle’s circumference.

The circumference of the circle is $4\pi$

What is the area of one of the triangles?

Answer

The radius of the circle is 2, from the equation circumference $=2r\pi$ . Each triangle is the same, and is equilateral, with side length of 2. The area of a triangle $=\frac{1}{2}bh$

To find the height of this triangle, we must divide it down the centerline, which will make two identical 30-60-90 triangles, each with a base of 1 and a hypotenuse of 2. Since these triangles are both right traingles (they have a 90 degree angle in them), we can use the Pythagorean Theorem to solve their height, which will be identical to the height of the equilateral triangle.

$a^{2}+b^{2}=c^{2}$

We know that the hypotenuse is 2 so $2^{2}=4$ . That's our $c^{2}$ solution. We know that the base is 1, and if you square 1, you get 1.

Now our formula looks like this: $a^{2}+1=4$ , so we're getting close to finding .

Let's subtract 1 from each side of that equation, in order to make things a bit simpler: $a^{2}+0=3 \rightarrow a^{2}=3$

Now let's apply the square root to each side of the equation, in order to change $a^{2}$ into : $a=\sqrt{3}$

Therefore, the height of our equilateral triangle is $\sqrt{3}$

To find the area of our equilateral triangle, we simply have to multiply half the base by the height: $\frac{2}{2}\sqrt{3}=1*\sqrt{3}=\sqrt{3}$

The area of our triangle is $\sqrt{3}$

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Question

What is the area of an equilateral triangle with a side length of 5?

Answer

Note that an equilateral triangle has equal sides and equal angles. The question gives us the length of the base, 5, but doesn't tell us the height.

If we split the triangle into two equal triangles, each has a base of 5/2 and a hypotenuse of 5.

Therefore we can use the Pythagorean Theorem to solve for the height:

$(\frac{5}{2})^2+b^2=5^2$

$\frac{25}{4}+b^2=25$

$b^2=\frac{75}{4}$

$b=\sqrt{\frac{3\times 25}{2\times 2}}=\frac{5\sqrt{3}}{2}$

Now we can find the area of the triangle:

$Area=\frac{1}{2}\times base\times height=\frac{1}{2}\times 5\times \frac{5\sqrt{3}}{2}=\frac{25\sqrt{3}}{4}$

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Question

What is the area of an equilateral triangle with sides of length ?

Answer

While you can very quickly compute the area of an equilateral triangle by using a shortcut formula, it is best to understand how to analyze a triangle like this for other problems. Let's consider this. The shortcut will be given below.

Recall that from any vertex of an equilateral triangle, you can drop a height that is a bisector of that vertex as well as a bisector of the correlative side. This gives you the following figure:

Notice that the small triangles within the larger triangle are both triangles. Therefore, you can create a ratio to help you find .

The ratio of to is the same as the ratio of to $\sqrt{3}$ .

As an equation, this is written:

$\frac{10}{h}=\frac{1}{\sqrt{3}}$

Solving for , we get: $h=10\sqrt{3}$

Now, the area of the triangle is merely $\frac{1}{2}bh$ . For our data, this is: $\frac{1}{2}2010\sqrt{3}$ or $100\sqrt{3}$ .

Notice that this is the same as $\frac{b^2\sqrt{3}}{4}$ . This is a shortcut formula for the area of equilateral triangles.

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Question

What is the area of an equilateral triangle with a perimeter of ?

Answer

Since an equilateral triangle is comprised of sides having equal length, we know that each side of this triangle must be $\frac{51}{3}$ or . While you can very quickly compute the area of an equilateral triangle by using a shortcut formula, it is best to understand how to analyze a triangle like this for other problems. Let's consider this. The shortcut will be given below.

Recall that from any vertex of an equilateral triangle, you can drop a height that is a bisector of that vertex as well as a bisector of the correlative side. This gives you the following figure:

Notice that the small triangles within the larger triangle are both triangles. Therefore, you can create a ratio to help you find .

The ratio of to is the same as the ratio of to $\sqrt{3}$ .

As an equation, this is written:

$\frac{8.5}{h}=\frac{1}{\sqrt{3}}$

Solving for , we get: $h=8.5\sqrt{3}$

Now, the area of the triangle is merely $\frac{1}{2}bh$ . For our data, this is: $\frac{1}{2}178.5\sqrt{3}$ or $72.25\sqrt{3}$ .

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Question

What is the area of a triangle with a circle inscribed inside of it, in terms of the circle's radius R?

Answer

Draw out 3 radii and 3 lines to the corners of each triangle, creating 6 30-60-90 triangles.

See that these 30-60-90 triangles can be used to find side length.

$S = R2\sqrt{3}$

Formula for side of equilateral triangle is

$\frac{\sqrt{3}}{4} * S^2$ .

Now substitute the new equation that is in terms of R in for S.

$=\frac{\sqrt{3}}{4}(R2\sqrt{3})^2$

$=\frac{\sqrt{3}}{4} 12 * R^{2}$

$= 3\sqrt{3}R^2$

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Question

What is the area of a triangle inscribed in a circle, in terms of the radius R of the circle?

Answer

Draw 3 radii, and then 3 new lines that bisect the radii. You get six 30-60-90 triangles.

These triangles can be used ot find a side length $S = R\sqrt{3}$ .

Using the formula for the area of an equilateral triangle in terms of its side, we get

$\frac{\sqrt{3}}{4}S^2$

$\frac{\sqrt{3}}{4}(R\sqrt3)^2 =$

$3\frac{\sqrt{3}}{4} * R^2$

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Question

In the figure below, right triangle has a hypotenuse of 6. If $TU=\sqrt{5x}$ and , find the perimeter of the triangle .

Answer

How do you find the perimeter of a right triangle?

There are three primary methods used to find the perimeter of a right triangle.

When side lengths are given, add them together.

Solve for a missing side using the Pythagorean theorem.

If we know side-angle-side information, solve for the missing side using the Law of Cosines.

Method 1:

This method will show you how to calculate the perimeter of a triangle when all sides lengths are known. Consider the following figure:

If we know the lengths of sides , , and , then we can simply add them together to find the perimeter of the triangle. It is important to note several things. First, we need to make sure that all the units given match one another. Second, when all the side lengths are known, then the perimeter formula may be used on all types of triangles (e.g. right, acute, obtuse, equilateral, isosceles, and scalene). The perimeter formula is written formally in the following format:

$\textup{Perimeter}=a+b+c$

Method 2:

In right triangles, we can calculate the perimeter of a triangle when we are provided only two sides. We can do this by using the Pythagorean theorem. Let's first discuss right triangles in a general sense. A right triangle is a triangle that has one $90^{\circ}$ angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the $90^{\circ}$ angle and they are labeled and . The side of the triangle that is opposite of the $90^{\circ}$ angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as .

If a triangle appears in this format, then we can use the Pythagorean theorem to solve for any missing side. This formula is written in the following manner:

We can rearrange it in a number of ways to solve for each of the sides of the triangle. Let's rearrange it to solve for the hypotenuse, .

Rearrange and take the square root of both sides.

$\sqrt{c^2}=\sqrt{a^2+b^2}$

Simplify.

$c=\sqrt{a^2+b^2}$

Now, let's use the Pythagorean theorem to solve for one of the legs, .

Subtract from both sides of the equation.

Take the square root of both sides.

$\sqrt{a^2}=\sqrt{c^2-b^2}$

Simplify.

$a=\sqrt{c^2-b^2}$

Last, let's use the Pythagorean theorem to solve for the adjacent leg, .

Subtract from both sides of the equation.

Take the square root of both sides.

$\sqrt{b^2}=\sqrt{c^2-a^2}$

Simplify.

$b=\sqrt{c^2-a^2}$

It is important to note that we can only use the following formulas to solve for the missing side of a right triangle when two other sides are known:

$c=\sqrt{a^2+b^2}$

$a=\sqrt{c^2-b^2}$

$b=\sqrt{c^2-a^2}$

After we find the missing side, we can use the perimeter formula to calculate the triangle's perimeter.

Method 3:

This method is the most complicated method and can only be used when we know two side lengths of a triangle as well as the measure of the angle that is between them. When we know side-angle-side (SAS) information, we can use the Law of Cosines to find the missing side. In order for this formula to accurately calculate the missing side we need to label the triangle in the following manner:

When the triangle is labeled in this way each side directly corresponds to the angle directly opposite of it. If we label our triangle carefully, then we can use the following formulas to find missing sides in any triangle given SAS information:

$a^2=b^2+c^2-2bc\cdot \cos A$

$b^2=a^2+c^2-2ac\cdot \cos B$

$c^2=a^2+b^2-2ab\cdot \cos C$

After, we calculate the right side of the equation, we need to take the square root of both sides in order to obtain the final side length of the missing side. Last, we need to use the perimeter formula to obtain the distance of the side lengths of the polygon.

Solution:

Now, that we have discussed the three methods used to calculate the perimeter of a triangle, we can use this information to solve the problem.

First, we need to use the Pythagorean theorem to solve for .

$x^2+(\sqrt{5x})^2=36$

Because we are dealing with a triangle, the only valid solution is because we can't have negative values.

After you have found , plug it in to find the perimeter. Remember to simplify all square roots!

$4+\sqrt{5\times4}+6$

$=10+2\sqrt5$

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Question

Find the perimeter of the triangle below.

Answer

How do you find the perimeter of a right triangle?

There are three primary methods used to find the perimeter of a right triangle.

When side lengths are given, add them together.

Solve for a missing side using the Pythagorean theorem.

If we know side-angle-side information, solve for the missing side using the Law of Cosines.

Method 1:

This method will show you how to calculate the perimeter of a triangle when all sides lengths are known. Consider the following figure:

If we know the lengths of sides , , and , then we can simply add them together to find the perimeter of the triangle. It is important to note several things. First, we need to make sure that all the units given match one another. Second, when all the side lengths are known, then the perimeter formula may be used on all types of triangles (e.g. right, acute, obtuse, equilateral, isosceles, and scalene). The perimeter formula is written formally in the following format:

$\textup{Perimeter}=a+b+c$

Method 2:

In right triangles, we can calculate the perimeter of a triangle when we are provided only two sides. We can do this by using the Pythagorean theorem. Let's first discuss right triangles in a general sense. A right triangle is a triangle that has one $90^{\circ}$ angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the $90^{\circ}$ angle and they are labeled and . The side of the triangle that is opposite of the $90^{\circ}$ angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as .

If a triangle appears in this format, then we can use the Pythagorean theorem to solve for any missing side. This formula is written in the following manner:

We can rearrange it in a number of ways to solve for each of the sides of the triangle. Let's rearrange it to solve for the hypotenuse, .

Rearrange and take the square root of both sides.

$\sqrt{c^2}=\sqrt{a^2+b^2}$

Simplify.

$c=\sqrt{a^2+b^2}$

Now, let's use the Pythagorean theorem to solve for one of the legs, .

Subtract from both sides of the equation.

Take the square root of both sides.

$\sqrt{a^2}=\sqrt{c^2-b^2}$

Simplify.

$a=\sqrt{c^2-b^2}$

Last, let's use the Pythagorean theorem to solve for the adjacent leg, .

Subtract from both sides of the equation.

Take the square root of both sides.

$\sqrt{b^2}=\sqrt{c^2-a^2}$

Simplify.

$b=\sqrt{c^2-a^2}$

It is important to note that we can only use the following formulas to solve for the missing side of a right triangle when two other sides are known:

$c=\sqrt{a^2+b^2}$

$a=\sqrt{c^2-b^2}$

$b=\sqrt{c^2-a^2}$

After we find the missing side, we can use the perimeter formula to calculate the triangle's perimeter.

Method 3:

This method is the most complicated method and can only be used when we know two side lengths of a triangle as well as the measure of the angle that is between them. When we know side-angle-side (SAS) information, we can use the Law of Cosines to find the missing side. In order for this formula to accurately calculate the missing side we need to label the triangle in the following manner:

When the triangle is labeled in this way each side directly corresponds to the angle directly opposite of it. If we label our triangle carefully, then we can use the following formulas to find missing sides in any triangle given SAS information:

$a^2=b^2+c^2-2bc\cdot \cos A$

$b^2=a^2+c^2-2ac\cdot \cos B$

$c^2=a^2+b^2-2ab\cdot \cos C$

After, we calculate the right side of the equation, we need to take the square root of both sides in order to obtain the final side length of the missing side. Last, we need to use the perimeter formula to obtain the distance of the side lengths of the polygon.

Solution:

Now, that we have discussed the three methods used to calculate the perimeter of a triangle, we can use this information to solve the problem. The perimeter of a triangle is simply the sum of its three sides. Our problem is that we only know two of the sides. The key for us is the fact that we have a right triangle (as indicated by the little box in the one angle). Knowing two sides of a right triangle and needing the third is a classic case for using the Pythagorean theorem. In simple (sort of), the Pythagorean theorem says that sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of its hypotenuse.

Every right triangle has three sides and a right angle. The side across from the right angle (also the longest) is called the hypotenuse. The other two sides are each called legs. That means in our triangle, the side with length 17 is the hypotenuse, while the one with length 8 and the one we need to find are each legs.

What the Pythagorean theorem tells us is that if we square the lengths of our two legs and add those two numbers together, we get the same number as when we square the length of our hypotenuse. Since we don't know the length of our second leg, we can identify it with the variable .

This allows us to create the following algebraic equation:

$x^{2}+8^2 = 17^2$

which simplified becomes

To solve this equation, we first need to get the variable by itself, which can be done by subtracting 64 from both sides, giving us

From here, we simply take the square root of both sides.

$x=\pm15$

Technically, would also be a square root of 225, but since a side of a triangle can only have a positive length, we'll stick with 15 as our answer.

But we aren't done yet. We now know the length of our missing side, but we still need to add the three side lengths together to find the perimeter.

Our answer is 40.

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Question

Given that two sides of a right triangle are and and the hypotenuse is unknown, find the perimeter of the triangle.

Answer

How do you find the perimeter of a right triangle?

There are three primary methods used to find the perimeter of a right triangle.

When side lengths are given, add them together.

Solve for a missing side using the Pythagorean theorem.

If we know side-angle-side information, solve for the missing side using the Law of Cosines.

Method 1:

This method will show you how to calculate the perimeter of a triangle when all sides lengths are known. Consider the following figure:

If we know the lengths of sides , , and , then we can simply add them together to find the perimeter of the triangle. It is important to note several things. First, we need to make sure that all the units given match one another. Second, when all the side lengths are known, then the perimeter formula may be used on all types of triangles (e.g. right, acute, obtuse, equilateral, isosceles, and scalene). The perimeter formula is written formally in the following format:

$\textup{Perimeter}=a+b+c$

Method 2:

In right triangles, we can calculate the perimeter of a triangle when we are provided only two sides. We can do this by using the Pythagorean theorem. Let's first discuss right triangles in a general sense. A right triangle is a triangle that has one $90^{\circ}$ angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the $90^{\circ}$ angle and they are labeled and . The side of the triangle that is opposite of the $90^{\circ}$ angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as .

If a triangle appears in this format, then we can use the Pythagorean theorem to solve for any missing side. This formula is written in the following manner:

We can rearrange it in a number of ways to solve for each of the sides of the triangle. Let's rearrange it to solve for the hypotenuse, .

Rearrange and take the square root of both sides.

$\sqrt{c^2}=\sqrt{a^2+b^2}$

Simplify.

$c=\sqrt{a^2+b^2}$

Now, let's use the Pythagorean theorem to solve for one of the legs, .

Subtract from both sides of the equation.

Take the square root of both sides.

$\sqrt{a^2}=\sqrt{c^2-b^2}$

Simplify.

$a=\sqrt{c^2-b^2}$

Last, let's use the Pythagorean theorem to solve for the adjacent leg, .

Subtract from both sides of the equation.

Take the square root of both sides.

$\sqrt{b^2}=\sqrt{c^2-a^2}$

Simplify.

$b=\sqrt{c^2-a^2}$

It is important to note that we can only use the following formulas to solve for the missing side of a right triangle when two other sides are known:

$c=\sqrt{a^2+b^2}$

$a=\sqrt{c^2-b^2}$

$b=\sqrt{c^2-a^2}$

After we find the missing side, we can use the perimeter formula to calculate the triangle's perimeter.

Method 3:

This method is the most complicated method and can only be used when we know two side lengths of a triangle as well as the measure of the angle that is between them. When we know side-angle-side (SAS) information, we can use the Law of Cosines to find the missing side. In order for this formula to accurately calculate the missing side we need to label the triangle in the following manner:

When the triangle is labeled in this way each side directly corresponds to the angle directly opposite of it. If we label our triangle carefully, then we can use the following formulas to find missing sides in any triangle given SAS information:

$a^2=b^2+c^2-2bc\cdot \cos A$

$b^2=a^2+c^2-2ac\cdot \cos B$

$c^2=a^2+b^2-2ab\cdot \cos C$

After, we calculate the right side of the equation, we need to take the square root of both sides in order to obtain the final side length of the missing side. Last, we need to use the perimeter formula to obtain the distance of the side lengths of the polygon.

Solution:

Now, that we have discussed the three methods used to calculate the perimeter of a triangle, we can use this information to solve the problem.

In order to calculate the perimeter we need to find the length of the hypotenuse using the Pythagorean theorem.

Rearrange.

$c=\sqrt{a^2+b^2}$

Substitute in known values.

$c=\sqrt{3^2+4^2}$

$c=\sqrt{9+16}$

$c=\sqrt{25}$

Now that we have found the missing side, we can substitute the values into the perimeter formula and solve.

$\textup{Perimeter}=a+b+c$

$\textup{Perimeter}=3+4+5$

$\textup{Perimeter}=12$

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Question

What is the height of an equilateral triangle with a side length of 8 in?

Answer

An equilateral triangle has three congruent sides, and is also an equiangular triangle with three congruent angles that each meansure 60 degrees.

To find the height we divide the triangle into two special 30 - 60 - 90 right triangles by drawing a line from one corner to the center of the opposite side. This segment will be the height, and will be opposite from one of the 60 degree angles and adjacent to a 30 degree angle. The special right triangle gives side ratios of , $x\sqrt{3}$ , and . The hypoteneuse, the side opposite the 90 degree angle, is the full length of one side of the triangle and is equal to . Using this information, we can find the lengths of each side fo the special triangle.

$8=2x\rightarrow 4=x\rightarrow 4\sqrt{3}=x\sqrt{3}$

The side with length $x\sqrt{3}$ will be the height (opposite the 60 degree angle). The height is $4\sqrt{3}$ inches.

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Question

Find the height of a triangle if all sides have a length of .

Answer

Draw a vertical line from the vertex. This will divide the equilateral triangle into two congruent right triangles. For the hypothenuse of one right triangle, the length will be . The base will have a dimension of . Use the Pythagorean Theorem to solve for the height, substituting in for , the length of the hypotenuse, and for either or , the length of the legs of the triangle:

$b=\sqrt3$

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Question

What is the height of an equilateral triangle with sides of length ?

Answer

Recall that from any vertex of an equilateral triangle, you can drop a height that is a bisector of that vertex as well as a bisector of the correlative side. This gives you the following figure:

Notice that the small triangles within the larger triangle are both triangles. Therefore, you can create a ratio to help you find .

The ratio of to is the same as the ratio of to $\sqrt{3}$ .

As an equation, this is written:

$\frac{6}{h}=\frac{1}{\sqrt{3}}$

Solving for , we get: $h=6\sqrt{3}$

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Question

The Triangle Perpendicular Bisector Theorem states that the perpendicular bisector of an equilateral triangle is also the triangle's height.

$\Delta ABC$ is an equilateral triangle with side length inches. What is the height of $\Delta ABC$ ?

Answer

To calculate the height of an equilateral triangle, first draw a perpendicular bisector for the triangle. By definition, this splits the opposite side from the vertex of the bisector in two, resulting in two line segments of length inches. Since it is perpendicular, we also know the angle of intersection is $90^{\circ}$ .

So, we have a new right triangle with two side lenghts and for the hypotenuse and short leg, respectively. The Pythagorean theorem takes over from here:

---> $x^2 = 48 = 4\sqrt{3}$

So, the height of our triangle is $4\sqrt{3}$ .

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Question

Triangle A and Triangle B are similar isosceles triangles. Triangle A's sides measure , , and . Two of the angles in Triangle A each measure $\small 83^o$ . Triangle B's sides measure , , and . What is the measure of the smallest angle in Triangle B?

Answer

Because the interior angles of a triangle add up to $\small 180^o$ , and two of Triangle A's interior angles measure $\small 83^o$ , we must simply add the two given angles and subtract from $\small 180^o$ to find the missing angle:

$\small 83^o+83^o=166^o$

$\small 180^o-166^o=14^o$

Therefore, the missing angle (and the smallest) from Triangle A measures $\small 14^o$ . If the two triangles are similar, their interior angles must be congruent, meaning that the smallest angle is Triangle B is also $\small 14^o$ .

The side measurements presented in the question are not needed to find the answer!

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Question

Triangle A and Triangle B are similar isosceles triangles. Triangle A has a base of and a height of . Triangle B has a base of . What is the length of Triangle B's two congruent sides?

Answer

We must first find the length of the congruent sides in Triangle A. We do this by setting up a right triangle with the base and the height, and using the Pythagorean Theorem to solve for the missing side ( $\small c$ ). Because the height line cuts the base in half, however, we must use for the length of the base's side in the equation instead of . This is illustrated in the figure below:

Using the base of $\small 3cm$ and the height of $\small 4cm$ , we use the Pythagorean Theorem to solve for $\small c$ :

$\small 3^2+4^2=c^2$

$\small 9+16=c^2$

$\small 25=c^2$

$\small c=5$

Therefore, the two congruent sides of Triangle A measure $\small 5cm$ ; however, the question asks for the two congruent sides of Triangle B. In similar triangles, the ratio of the corresponding sides must be equal. We know that the base of Triangle A is and the base of Triangle B is . We then set up a cross-multiplication using the ratio of the two bases and the ratio of $\small c$ to the side we're trying to find ( $\small x$ ), as follows:

$\small \frac{6}{12}=\frac{5}{x}$

$\small 6x=60$

$\small x=10$

Therefore, the length of the congruent sides of Triangle B is .

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