# SAT Math : Plane Geometry

## Example Questions

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

Figure NOT drawn to scale.

is a right triangle with altitude . What percent of  is shaded in?

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 and to the large triangle. From the Pythagorean Theorem, the hypotenuse of  has length

.

The similarity ratio of  to  is the ratio of the lengths of the hypotenuses:

The ratio of the areas of two similar triangles is the square of their similarity ratio, which here is

Therefore, the area of  is

the overall area of . This makes  the closest response.

### Example Question #11 : Triangles

The perimeter of a right triangle is 40 units. If the lengths of the sides are , , and  units, then what is the area of the triangle?

Explanation:

Because the perimeter is equal to the sum of the lengths of the three sides of a triangle, we can add the three expressions for the lengths and set them equal to 40.

Perimeter:

Simplify the x terms.

Simplify the constants.

Subtract 8 from both sides.

Divide by 4

One side is 8.

The second side is

.

The third side is

.

Thus, the sides of the triangle are 8, 15, and 17.

The question asks us for the area of the triangle, which is given by the formula (1/2)bh. We are told it is a right triangle, so we can use one of the legs as the base, and the other leg as the height, since the legs will intersect at right angles. The legs of the right triangle must be the smallest sides (the longest must be the hypotenuse), which in this case are 8 and 15. So, let's assume that 8 is the base and 15 is the height.

The area of a triangle is (1/2)bh. We can substitute 8 and 15 for b and h.

.

The answer is 60 units squared.

### Example Question #2 : How To Find The Area Of A Right Triangle

The vertices of a right triangle on the coordinate axes are at the origin, , and . Give the area of the triangle.

Explanation:

The triangle in question can be drawn as the following:

The lengths of the legs of the triangle are 12, the distance from the origin to , and 8, the distance from the origin to . The area of a right triangle is equal to half the product of the lengths of the legs, so set  in the formula:

### Example Question #123 : Plane Geometry

Three points in the xy-coordinate system form a triangle.

The points are .

What is the perimeter of the triangle?

Explanation:

Drawing points gives sides of a right triangle of 4, 5, and an unknown hypotenuse.

Using the pythagorean theorem we find that the hypotenuse is .

### Example Question #124 : Plane Geometry

Based on the information given above, what is the perimeter of triangle ABC?

Explanation:

Consult the diagram above while reading the solution. Because of what we know about supplementary angles, we can fill in the inner values of the triangle. Angles A and B can be found by the following reductions:

A + 120 = 180; A = 60

B + 150 = 180; B = 30

Since we know A + B + C = 180 and have the values of A and B, we know:

60 + 30 + C = 180; C = 90

This gives us a 30:60:90 triangle. Now, since 17.5 is across from the 30° angle, we know that the other two sides will have to be √3 and 2 times 17.5; therefore, our perimeter will be as follows:

### Example Question #11 : Triangles

Give the perimeter of the provided triangle.

Explanation:

The figure shows a right triangle. The acute angles of a right triangle have measures whose sum is , so

Substituting  for :

This makes  a 45-45-90 triangle.

By the 45-45-90 Triangle Theorem, legs  and  are of the same length, so

.

Also by the 45-45-90 Triangle Theorem, the length of hypotenuse is equal to that of leg  multiplied by . Therefore,

.

The perimeter of the triangle is

### Example Question #2 : How To Find The Perimeter Of A Right Triangle

What is the perimeter of the triangle above?

Explanation:

The figure shows a right triangle. The acute angles of a right triangle have measures whose sum is , so

Substituting  for :

This makes  a 45-45-90 triangle. By the 45-45-90 Triangle Theorem, the length of leg  is equal to that of hypotenuse , the length of which is 12, divided by . Therefore,

Rationalize the denominator by multiplying both halves of the fraction by :

By the same reasoning, .

The perimeter of the triangle is

### Example Question #1 : Apply The Pythagorean Theorem To Find The Distance Between Two Points In A Coordinate System: Ccss.Math.Content.8.G.B.8

If  and , how long is side ?

Not enough information to solve

Explanation:

This problem is solved using the Pythagorean theorem  .  In this formula  and  are the legs of the right triangle while  is the hypotenuse.

Using the labels of our triangle we have:

### Example Question #1 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

If one of the short sides of a 45-45-90 triangle equals 5, how long is the hypotenuse?

√15

5

π

√10

5√2

5√2

Explanation:

Using the Pythagorean theorem, x2 + y2 = h2. And since it is a 45-45-90 triangle the two short sides are equal. Therefore 52 + 52 = h2 .  Multiplied out 25 + 25 = h2.

Therefore h2 = 50, so h = √50 = √2 * √25 or 5√2.

### Example Question #1 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

The height of a right circular cylinder is 10 inches and the diameter of its base is 6 inches. What is the distance from a point on the edge of the base to the center of the entire cylinder?

√(34)

4π/5

√(43)/2

3π/4

√(34)

Explanation:

The best thing to do here is to draw diagram and draw the appropiate triangle for what is being asked. It does not matter where you place your point on the base because any point will produce the same result.  We know that the center of the base of the cylinder is 3 inches away from the base (6/2). We also know that the center of the cylinder is 5 inches from the base of the cylinder (10/2). So we have a right triangle with a height of 5 inches and a base of 3 inches. So using the Pythagorean Theorem 3+ 5= c2. 34 = c2, c = √(34).