Geometry - Acute / Obtuse Isosceles Triangles

An isosceles triangle has a perimeter of . If the base of the triangle is two less than two times the length of each leg, what is the height of the triangle?

The height of the triangle cannot be determined with the given information.

Explanation

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be .

Use the information given about the perimeter to solve for .

Plug this value in to find the length of the base.

$\text{Length of Base}=2(6)-2=10$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{6^2-5^2}=\sqrt{11}=3.32$

Make sure to round to places after the decimal.

An isosceles triangle has a perimeter of . If the base of the triangle is two less than two times the length of each leg, what is the height of the triangle?

The height of the triangle cannot be determined with the given information.

Explanation

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be .

Use the information given about the perimeter to solve for .

Plug this value in to find the length of the base.

$\text{Length of Base}=2(6)-2=10$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{6^2-5^2}=\sqrt{11}=3.32$

Make sure to round to places after the decimal.

An isosceles triangle is placed in a circle as shown by the figure below.

If diameter of the circle is , find the area of the shaded region.

Explanation

From the given image, you should notice that the base of the triangle is also the diameter of the circle. In addition, the height of the triangle is also the radius of the circle.

Thus, we can find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(2 \times 1)=1$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of Circle}=\pi\times 1^2= \pi$

To find the area of the shaded region, subtract the two areas.

$\text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$

$\text{Area of Shaded Region}=\pi-1=2.14$

Make sure to round to places after the decimal.

If a triangle has side lengths of and , which of the following can be a length of the third side?

Explanation

The triangle inequality theorem states that the sum of the lengths of any two sides must be greater than the length of the third side. The relationship can be represented by the following inequalities:

The side length of is the only choice that fits this criteria:

$\bigtriangleup ABC$ is an equilateral triangle; $\bigtriangleup DEF$ is an equiangular triangle.

True or false: From the given information, it follows that $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

True

False

Explanation

As we are establishing whether or not $\bigtriangleup ABC \sim \bigtriangleup DEF$ , then , , and correspond respectively to , , and .

A triangle is equilateral (having three sides of the same length) if and only if it is also equiangular (having three angles of the same measure, each of which is $60^{\circ }$ ). It follows that all angles of both triangles measure $60^{\circ }$ .

Specifically, $\angle A \cong \angle D$ and $\angle B \cong \angle E$ , making two pairs of corresponding angles congruent. By the Angle-Angle Similarity Postulate, it follows that $\bigtriangleup ABC \sim \bigtriangleup DEF$ , making the correct answer "true".

$\bigtriangleup ABC$ is an equilateral triangle; $\bigtriangleup DEF$ is an equiangular triangle.

True or false: From the given information, it follows that $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

True

False

Explanation

As we are establishing whether or not $\bigtriangleup ABC \sim \bigtriangleup DEF$ , then , , and correspond respectively to , , and .

An isosceles triangle is placed in a circle as shown by the figure below.

If diameter of the circle is , find the area of the shaded region.

Explanation

From the given image, you should notice that the base of the triangle is also the diameter of the circle. In addition, the height of the triangle is also the radius of the circle.

Thus, we can find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(2 \times 1)=1$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of Circle}=\pi\times 1^2= \pi$

To find the area of the shaded region, subtract the two areas.

$\text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$

$\text{Area of Shaded Region}=\pi-1=2.14$

Make sure to round to places after the decimal.

If a triangle has side lengths of and , which of the following can be a length of the third side?

Explanation

The side length of is the only choice that fits this criteria:

An isosceles triangle has a perimeter of . If the base of the triangle is less than three times the length of a leg, what is the height of the triangle?

Explanation

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be .

Use the information given about the perimeter to solve for .

Plug this value in to find the length of the base.

$\text{Length of Base}=3(10)-18=12$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{10^2-6^2}=\sqrt{64}=8$

Make sure to round to places after the decimal.

Parallel 2

Refer to the above diagram. $\angle ABV \cong \angle DCV$ .

True or false: From the information given, it follows that $\bigtriangleup AVB \sim \bigtriangleup CVD$ .

False

True

Explanation

The given information is actually inconclusive.

By the Angle-Angle Similarity Postulate, if two pairs of corresponding angles of a triangle are congruent, the triangles themselves are similar. Therefore, we seek to prove two of the following three angle congruence statements:

$\angle AVB \cong \angle CVD$

$\angle ABV \cong \angle CDV$

$\angle BAV \cong \angle DCV$

$\angle AVB$ and $\angle CVD$ are a pair of vertical angles, having the same vertex and having sides opposite each other. As such, $\angle AVB \cong \angle CVD$ .

$\angle ABV \cong \angle DCV$ , but this is not one of the statements we need to prove. Also, without further information - for example, whether and are parallel, which is not given to us - we have no way to prove either of the other two necessary statements.

The correct response is "false".

Acute / Obtuse Isosceles Triangles

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