# ISEE Upper Level Math : Triangles

## Example Questions

### Example Question #31 : Triangles

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? Possible Answers: Correct answer: Explanation: 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.

### Example Question #71 : Plane Geometry

Which of the following is true about a triangle with two angles that measure  and ?

This triangle is obtuse.

This is a right isosceles triangle.

This triangle is scalene and right.

This triangle is scalene.

This is an equilateral triangle.

This is a right isosceles triangle.

Explanation:

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.

### Example Question #33 : Triangles

What is the value of x in a right triangle if the two acute angles are equal to 5x and 25x?

Explanation:

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:

### Example Question #1 : How To Find The Area Of A Right Triangle

One angle of a right triangle measures 45 degrees, and the hypotenuse measures 8 centimeters. Give the area of the triangle.

Explanation:

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.

### Example Question #35 : Triangles

The legs of a right triangle measure  and . What is its perimeter?

Explanation:

The hypotenuse of the triangle can be calculated using the Pythagorean Theorem. Set :

### Example Question #1 : How To Find The Area Of A Right Triangle

Figure NOT drawn to scale

is a right triangle with altitude . Give the ratio of the area of to that of .

Explanation:

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 .

### Example Question #72 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Find the area of a right triangle with a base of 7cm and a height of 20cm.

Explanation:

To find the area of a right triangle, we will use the following formula:

where b is the base and 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

### Example Question #1 : Acute / Obtuse Triangles

NOTE: Figures NOT drawn to scale.

Refer to the above two triangles.

What is ?

Insufficient information is given to answer the question.

Explanation:

Corresponding sides of similar triangle are proportional, so if

, then

Substitute the known sidelengths, then solve for :

### Example Question #81 : Geometry

What is the perimeter of  ?

It cannot be determined from the information given.

Explanation:

By definition, since, , side lengths are in proportion.

So,

The perimeter of  is

.

What is  ?