How to find the height of a right triangle - ISEE Upper Level Quantitative Reasoning
Card 1 of 24
Refer to the above figure. Evaluate the length of 
in terms of 
.
Refer to the above figure. Evaluate the length of
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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
Tap to reveal answer
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,
Tap to reveal answer
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
Tap to reveal answer
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?
Tap to reveal answer
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
Tap to reveal answer
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 figure. Evaluate the length of in terms of .
Refer to the above figure. Evaluate the length of
Tap to reveal answer
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
Tap to reveal answer
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,
Tap to reveal answer
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
Tap to reveal answer
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.
← Didn't Know|Knew It →
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?
Tap to reveal answer
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.
← Didn't Know|Knew It →
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
Tap to reveal answer
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 figure. Evaluate the length of in terms of .
Refer to the above figure. Evaluate the length of
Tap to reveal answer
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,
← Didn't Know|Knew It →
Refer to the above figure. Evaluate the length of in terms of .
Refer to the above figure. Evaluate the length of
Tap to reveal answer
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,
← Didn't Know|Knew It →
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,
Tap to reveal answer
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
Tap to reveal answer
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.
← Didn't Know|Knew It →
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?
Tap to reveal answer
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.
← Didn't Know|Knew It →
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
Tap to reveal answer
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.
← Didn't Know|Knew It →
Refer to the above figure. Evaluate the length of in terms of .
Refer to the above figure. Evaluate the length of
Tap to reveal answer
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,
← Didn't Know|Knew It →
Refer to the above figure. Evaluate the length of in terms of .
Refer to the above figure. Evaluate the length of
Tap to reveal answer
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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