Triangles - ISEE Upper Level Quantitative Reasoning
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One of the base angles of an isosceles triangle is 
. Give the measure of the vertex angle.
One of the base angles of an isosceles triangle is
The base angles of an isosceles triangle are always equal. Therefore both base angles are .
Let 
the measure of the third angle. Since the sum of the angles of a triangle is 
, we can solve accordingly:
The base angles of an isosceles triangle are always equal. Therefore both base angles are
Let
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Two sides of an isosceles triangle have lengths 3 feet and 4 feet. Which of the following could be the length of the third side?
Two sides of an isosceles triangle have lengths 3 feet and 4 feet. Which of the following could be the length of the third side?
An isosceles triangle, by definition, has two sides of equal length. Having the third side measure either 3 feet or 4 feet would make the triangle meet this criterion.
3 feet is equal to 
inches, and 4 feet is equal to 
inches. We choose 36 inches, since that, but not 48 inches, is a choice.
An isosceles triangle, by definition, has two sides of equal length. Having the third side measure either 3 feet or 4 feet would make the triangle meet this criterion.
3 feet is equal to
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The triangles are similar. Solve for 
.
The triangles are similar. Solve for
Because the triangles are similar, proportions can be used to solve for the length of the side:
Cross-multiply:
Because the triangles are similar, proportions can be used to solve for the length of the side:
Cross-multiply:
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Which of the following is true about a triangle with two angles that measure 
and 
?
Which of the following is true about a triangle with two angles that measure
The sum of the two given angles is 90 degrees, which means that the third angle should be a right angle (90 degrees). We also know that two of the angles are equal. Therefore, the triangle is right and isosceles.
The sum of the two given angles is 90 degrees, which means that the third angle should be a right angle (90 degrees). We also know that two of the angles are equal. Therefore, the triangle is right and isosceles.
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What is the value of x in a right triangle if the two acute angles are equal to 5x and 25x?
What is the value of x in a right triangle if the two acute angles are equal to 5x and 25x?
In a right triangle, there is one right angle of 90 degrees, while the two acute angles add up to 90 degrees.
Given that the two acute angles are equal to 5x and 25x, the value of x can be calculated with the equation below:
In a right triangle, there is one right angle of 90 degrees, while the two acute angles add up to 90 degrees.
Given that the two acute angles are equal to 5x and 25x, the value of x can be calculated with the equation below:
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is a right angle; 
, 
.
Which is the greater quantity?
(a) 
(b) 
Which is the greater quantity?
(a)
(b)
. Corresponding angles of similar triangles are congruent, so since 
is a right angle, so is 
.
The hypotenuse 
of 
is twice as long as leg 
; by the 
Theorem, 
. Again, by similiarity,
.
The hypotenuse
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One angle of a right triangle measures 45 degrees, and the hypotenuse measures 8 centimeters. Give the area of the triangle.
One angle of a right triangle measures 45 degrees, and the hypotenuse measures 8 centimeters. Give the area of the triangle.
This triangle has two angles of 45 and 90 degrees, so the third angle must measure 45 degrees; this is therefore an isosceles right triangle.
By the Pythagorean Theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Let 
hypotenuse and 
side length.
We can then plug this side length into the formula for area.
This triangle has two angles of 45 and 90 degrees, so the third angle must measure 45 degrees; this is therefore an isosceles right triangle.
By the Pythagorean Theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Let
We can then plug this side length into the formula for area.
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The legs of a right triangle measure 
and 
. What is its perimeter?
The legs of a right triangle measure
The hypotenuse of the triangle can be calculated using the Pythagorean Theorem. Set 
:
Add the three sidelengths:
The hypotenuse of the triangle can be calculated using the Pythagorean Theorem. Set
Add the three sidelengths:
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Figure NOT drawn to scale
is a right triangle with altitude 
. Give the ratio of the area of 
to that of 
.
Figure NOT drawn to scale
The altitude of a right triangle from the vertex of its right angle - which, here, is 
- divides the triangle into two triangles similar to each other. The ratio of the hypotenuse of the white triangle to that of the gray triangle (which are corresponding sides) is
,
making this the similarity ratio. The ratio of the areas of two similar triangles is the square of their similarity ratio, which here is
, or 
.
The altitude of a right triangle from the vertex of its right angle - which, here, is
making this the similarity ratio. The ratio of the areas of two similar triangles is the square of their similarity ratio, which here is
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Find the area of a right triangle with a base of 7cm and a height of 20cm.
Find the area of a right triangle with a base of 7cm and a height of 20cm.
To find the area of a right triangle, we will use the following formula:
where b is the base and h is the height of the triangle.
We know the base of the triangle is 7cm. We know the height of the triangle is 20cm. Knowing this, we can substitute into the formula. We get
To find the area of a right triangle, we will use the following formula:
where b is the base and h is the height of the triangle.
We know the base of the triangle is 7cm. We know the height of the triangle is 20cm. Knowing this, we can substitute into the formula. We get
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Refer to the above figure. Evaluate the length of 
in terms of 
.
Refer to the above figure. Evaluate the length of
The altitude of a right triangle to its hypotenuse divides the triangle into two smaller trangles similar to each other and to the large triangle.
Therefore,
and, consequently,
,
or, equivalently,
by the Pythagorean Theorem, so
.
The altitude of a right triangle to its hypotenuse divides the triangle into two smaller trangles similar to each other and to the large triangle.
Therefore,
and, consequently,
or, equivalently,
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Refer to the above figure. Evaluate the length of 
in terms of 
.
Refer to the above figure. Evaluate the length of
The height of a right triangle from the vertex of its right angle is the geometric mean - in this case, the square root of the product - of the lengths of the two segments of the hypotenuse that it forms. Therefore,
The height of a right triangle from the vertex of its right angle is the geometric mean - in this case, the square root of the product - of the lengths of the two segments of the hypotenuse that it forms. Therefore,
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Note: Figure NOT drawn to scale.
In the above right triangle, 
. Give the length of 
.
Note: Figure NOT drawn to scale.
In the above right triangle,
The two triangles formed by an altitude from the vertex of a right triangle are similar to each other and the large triangle, so all three are 30-60-90 triangles. Take advantage of this, applying it twice.
Looking at 
. By the 30-60-90 Theorem, the shorter leg of a hypotentuse measures half that of the hypotenuse. 
.
Now, look at 
. By the same theorem,
and
The two triangles formed by an altitude from the vertex of a right triangle are similar to each other and the large triangle, so all three are 30-60-90 triangles. Take advantage of this, applying it twice.
Looking at
Now, look at
and
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Note: Figure NOT drawn to scale.
In the above figure, 
.
Which of the following comes closest to the length of 
?
Note: Figure NOT drawn to scale.
In the above figure,
Which of the following comes closest to the length of
and 
, so by the Pythagorean Theorem,
Because 
is the altitude from the vertex 
of 
,
.
Therefore,
Also,
For similar reasons,
.
Therefore,
Of the choices given, 60 comes closest.
Because
Therefore,
Also,
For similar reasons,
Therefore,
Of the choices given, 60 comes closest.
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Note: Figure NOT drawn to scale.
Refer to the above map. A farmer owns a triangular plot of land flanked by the three highways as shown. The farmer uses Highway 2 frequently; however, he can only access it by driving five miles north on Highway 32 or twelve miles east on Highway 100.
He wants to build a dirt road that directly accesses Highway 2. He figures that it will cost $1,500 per tenth of a mile to construct the road. By his estimate, which answer will come closest to the total cost of the shortest possible road?
Note: Figure NOT drawn to scale.
Refer to the above map. A farmer owns a triangular plot of land flanked by the three highways as shown. The farmer uses Highway 2 frequently; however, he can only access it by driving five miles north on Highway 32 or twelve miles east on Highway 100.
He wants to build a dirt road that directly accesses Highway 2. He figures that it will cost $1,500 per tenth of a mile to construct the road. By his estimate, which answer will come closest to the total cost of the shortest possible road?
The three roads form the legs and the hypotenuse of a right triangle with legs 5 and 12 miles; by the Pythagorean Theorem, the hypotenuse is
miles.
To find the length of the shortest possible road, which must be perpendicular to Highway 2, it should be observed that this road serves as an altitude from the vertex of the triangle to the hypotenuse.
The area of a right triangle can be calculated two ways - by taking half the product of the lengths of the legs or by taking half the product of the length of the altitude (height) and that of the hypotenuse. Setting 
as the height, we can solve for 
in the equation:
miles.
This is 46 tenths of a mile, so the approximate cost of the road in dollars will be
Among the five choices, $70,000 comes closest.
The three roads form the legs and the hypotenuse of a right triangle with legs 5 and 12 miles; by the Pythagorean Theorem, the hypotenuse is
To find the length of the shortest possible road, which must be perpendicular to Highway 2, it should be observed that this road serves as an altitude from the vertex of the triangle to the hypotenuse.
The area of a right triangle can be calculated two ways - by taking half the product of the lengths of the legs or by taking half the product of the length of the altitude (height) and that of the hypotenuse. Setting
This is 46 tenths of a mile, so the approximate cost of the road in dollars will be
Among the five choices, $70,000 comes closest.
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Note: Figure NOT drawn to scale.
In the above figure, 
.
Which of the following comes closest to the length of 
?
Note: Figure NOT drawn to scale.
In the above figure,
Which of the following comes closest to the length of
and 
, so by the Pythagorean Theorem,
Because 
is the altitude from the vertex 
of 
,
.
Therefore,
Also,
For similar reasons,
.
Therefore,
.
Of the choices given, 75 comes closest.
Because
Therefore,
Also,
For similar reasons,
Therefore,
Of the choices given, 75 comes closest.
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Refer to the above diagram, which depicts a right triangle. What is the value of 
?
Refer to the above diagram, which depicts a right triangle. What is the value of
By the Pythagorean Theorem, which says 
. 
being the hypotenuse, or 
in this problem.
Simply
By the Pythagorean Theorem, which says
Simply
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If a right triangle has a base of 
and a height of 
, what is the length of the hypotenuse?
If a right triangle has a base of
To solve this problem, we must utilize the Pythagorean Theorom, which states that:
We know that the base is 
, so we can substitute 
in for 
. We also know that the height is 
, so we can substitute 
in for 
.
Next we evaluate the exponents:
Now we add them together:
Then, 
.
is not a perfect square, so we simply write the square root as 
.
To solve this problem, we must utilize the Pythagorean Theorom, which states that:
We know that the base is
Next we evaluate the exponents:
Now we add them together:
Then,
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If a right triangle has a base of 
and a height of 
, what is the length of the hypotenuse?
If a right triangle has a base of
To solve this problem, we are going to use the Pythagorean Theorom, which states that 
.
We know that this particular right triangle has a base of 
, which can be substituted for 
, and a height of 
, which can be substituted for 
. If we rewrite the theorom using these numbers, we get:
Next, we evaluate the expoenents:
Then, 
.
To solve for 
, we must find the square root of 
. Since this is not a perfect square, our answer is simply 
.
To solve this problem, we are going to use the Pythagorean Theorom, which states that
We know that this particular right triangle has a base of
Next, we evaluate the expoenents:
Then,
To solve for
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What is the hypotenuse of a right triangle with sides 5 and 8?
What is the hypotenuse of a right triangle with sides 5 and 8?
According to the Pythagorean Theorem, the equation for the hypotenuse of a right triangle is 
. Plugging in the sides, we get 
. Solving for 
, we find that the hypotenuse is 
:
According to the Pythagorean Theorem, the equation for the hypotenuse of a right triangle is
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