ISEE Upper Level Math : How to find the height of a right triangle
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Example Questions
Example Question #1 : How To Find The Height Of A Right Triangle
Note: Figure NOT drawn to scale.
In the above figure, 
Which of the following comes closest to the length of 
Because 
Therefore,
Also,
For similar reasons,
Therefore,
Of the choices given, 60 comes closest.
Example Question #61 : Geometry
Note: Figure NOT drawn to scale.
In the above figure, 
Which of the following comes closest to the length of 
Because 
Therefore,
Also,
For similar reasons,
Therefore,
Of the choices given, 75 comes closest.
Example Question #1 : How To Find The Height Of A Right Triangle
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,
Example Question #4 : How To Find The Height Of A Right Triangle
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,
Example Question #5 : How To Find The Height Of A Right Triangle
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 
Now, look at 
and
Example Question #1 : How To Find The Height Of A Right Triangle
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
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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