Apply the Distance Formula to Determine Perimeter & Area of Triangles and Rectangles

Apply the distance formula to determine perimeter & area of triangles and rectangles.

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Printed: June 20, 2016 www.ck12.org Chapter 1. Apply the distance formula to determine perimeter & area of triangles and rectangles.

CHAPTER 1 Apply the distance formula to determine perimeter & area of triangles and rectangles.

Example1: Find the perimeter of the triangle below.

Solution: Use the distance formula to ﬁnd the length of each side.

p √ AB = 22 + 32 = 13 ≈ 3.61 un p √ BC = 12 + 22 = 5 ≈ 2.24 un p √ √ AC = 22 + 42 = 20 = 2 5 ≈ 4.47 un

The perimeter of√ the triangle√ is the√ distance√ around√ the triangle, which is the sum of the lengths of the three sides. The perimeter is 13 + 5 + 2 5 = 13 + 3 5 ≈ 10.31 units. 1 3 Next, you need to ﬁnd where this line intersects AC. The equation of the line that contains AC is y = 2 x + 2 . Solve the system of equations to ﬁnd the point of intersection. 1 3 y = −2x + 6 and y = 2 x + 2

1 3 −2x + 6 = x + 2 2 9 5 = x 2 2 x = 1.8 y = −2(1.8) + 6 y = 2.4

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The point of intersection is (1.8,2.4).

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Next, you need to ﬁnd the length of the height, which is the distance from (1.8,2.4) to (1,4). q h = (1.8 − 1)2 + (2.4 − 4)2 = 1.79 un Now that you know the lengths of the base and the height, you can ﬁnd the area of the triangle. The area of the triangle is:

bh A = 2 (4.47)(1.79) A = 2 A ≈ 4 un2

Vocabulary

The distance formula is a formula that helps calculate the distance between two points or the length of a line segment q 2 2 in the coordinate plane. The distance between (x1,y1) and (x2,y2) is (x2 − x1) + (y2 − y1) .

Guided Practice

1. Find the perimeter of the rectangle below.

2. Find the area of the rectangle from #1. 3. Find the distance between (−16,4) and (312,211). How does this calculation help to show why having the distance formula is helpful?

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Answers: 1. Because the polygon is a rectangle, its opposite sides are the same length. Use the distance formula to ﬁnd the length of two of the sides.

p √ AB = 32 + 12 = 10 p √ √ BC = 62 + 22 = 40 = 2 10

The perimeter of the rectangle is:

√ √ √ √ √ P = 10 + 2 10 + 10 + 2 10 = 6 10 un

2. To ﬁnd the area of the rectangle, use the formula A = bh. The base and the height lengths were found in #1.

√ √ A = 10 2 10 A = 20 un2

3. Let (x1,y1)= (−16,4) and (x2,y2) = (312,211).

q d = (312 − (−16))2 + (211 − 4)2 ≈ 387.01 un

Because these two points are so far apart, it would have been unrealistic to plot them and try to ﬁnd the distance between them by drawing in a right triangle and using the Pythagorean Theorem. The distance formula makes these types of calculations much quicker.

Practice

Find the distance between each pair of points. 1. (7,11) and (4,23). 12.37 units 2. (19,56) and (−21,45). 41.48 units 3. (−11,91) and (89,16). 125 units

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For #4-#5, use the rectangle below.

4. Find the perimeter of the rectangle. 14.14 units 5. Find the area of the rectangle. 12 units2 For #6-#7, use the triangle below.

6. Find the perimeter of the triangle. 10.94 units

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For #7-#8, use the rectangle below.

7. Find the perimeter of the rectangle. 13.22 units 8. Find the area of the rectangle. 9.56 units2 For #9-#10, use the right triangle below.

9. Find the perimeter of the triangle. 14.81 units 10. Find the area of the triangle. 8 units2

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References

1. . . CC BY-NC-SA 2. . . CC BY-NC-SA 3. . . CC BY-NC-SA 4. . . CC BY-NC-SA 5. . . CC BY-NC-SA 6. . . CC BY-NC-SA 7. . . CC BY-NC-SA