## SAT SUBJECT TEST MATH LEVEL 1

## PLANE GEOMETRY

**CHAPTER 9 ****Triangles**

**PERIMETER AND AREA**

The ** perimeter** of a triangle is the sum of the lengths of the three sides.

**EXAMPLE 6:** To find the perimeter of an equilateral triangle whose height is 12, note that the height divides the triangle into two 30-60-90 right triangles. In the figure below, by KEY FACT H8,

So the perimeter is 3*s* =24 .

**Key Fact H9**

**TRIANGLE INEQUALITY**

• **The sum of the lengths of any two sides of a triangle is greater than the length of the third side.**

• **The difference of the lengths of any two sides of a triangle is less than the length of the third side.**

**EXAMPLE 7:** A teacher asked her class to draw triangles in which the lengths of two of the sides were 6 inches and 7 inches and the length of the third side was also a whole number of inches. To determine how many different triangles the class could draw, note that if *x* represents the length of the third side, then by KEY FACT H9, 6 + 7 > *x*, so *x* < 13. Also, 7 – 6 < *x*, so *x* > 1. So *x* could have 11 different values: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. The perimeters of the triangles are the integers from 15 to 25. The following diagram illustrates four of the triangles the class could have drawn.

Frequently, questions on the Math 1 test require you to calculate the ** area** of a triangle.

**Key Fact H10**

**The area of a triangle is given by** ** bh, where b and h are the lengths of the base and height, respectively.**

**(1)** *Any* side of the triangle can be taken as the base.

**(2)** The ** height** (which is also called an

**) is a line segment drawn perpendicular to the base from the opposite vertex.**

*altitude***(3)** In a right triangle, either leg can be the base and the other the height.

**(4)** If one endpoint of the base is the vertex of an obtuse angle, then the height will be outside the triangle.

**(5)** In each figure at the top of this page

• If is the base, is the height.

• If is the base, is the height.

• If is the base, is the height.

**TIP**

The height can be outside the triangle. See in the diagram shown below.

**EXAMPLE 8:** To find the area of equilateral triangle *PQR*, below, whose sides are 12, draw in altitude . *PQS* is a 30-60-90 right triangle, and so by KEY FACT H8, *QS* = 6 and *PS* =6. Finally, the area of

Replacing 12 by *s* in Example 8 yields a useful formula.

**Key Fact H11**

**if A represents the area of an equilateral triangle with side s, then**

**.**

**Smart Strategy**

Learning this formula for the area of an equilateral triangle can save you time.