Question

find the approximate perimeter of △abc plotted below. a(2,7), c(6,7), b(-4,-3) choose 1 answer: a 24, b 25.7, c 28.1, d 29.8

Explanation:

Step1: Find length of AC

Points A(2,7) and C(6,7) have same y - coordinate. So, length \( AC=\vert6 - 2\vert=4\).

Step2: Find length of AB

Using distance formula \( d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \) for A(2,7) and B(-4,-3).
\( x_1 = 2,y_1 = 7,x_2=-4,y_2=-3 \)
\( AB=\sqrt{(-4 - 2)^2+(-3 - 7)^2}=\sqrt{(-6)^2+(-10)^2}=\sqrt{36 + 100}=\sqrt{136}\approx11.66 \)

Step3: Find length of BC

Using distance formula for B(-4,-3) and C(6,7).
\( x_1=-4,y_1=-3,x_2 = 6,y_2 = 7 \)
\( BC=\sqrt{(6+4)^2+(7 + 3)^2}=\sqrt{10^2+10^2}=\sqrt{100 + 100}=\sqrt{200}\approx14.14 \)

Step4: Find perimeter

Perimeter \(=AC + AB+BC=4 + 11.66+14.14 = 29.8\)? Wait, no, wait. Wait, AC is 4, AB: \(\sqrt{(-4 - 2)^2+(-3 - 7)^2}=\sqrt{36 + 100}=\sqrt{136}\approx11.66\), BC: \(\sqrt{(6+4)^2+(7 + 3)^2}=\sqrt{100 + 100}=\sqrt{200}\approx14.14\). Wait, 4+11.66 = 15.66+14.14 = 29.8? But wait, maybe I made a mistake. Wait, A(2,7), B(-4,-3): difference in x: - 6, difference in y: - 10. So squared: 36 and 100, sum 136, square root ~11.66. B(-4,-3) and C(6,7): difference in x:10, difference in y:10, squared 100 and 100, sum 200, square root ~14.14. AC is 4. So 4 + 11.66+14.14 = 29.8? But wait the options have D as 29.8. Wait, but let me check again. Wait, maybe I miscalculated AB. Wait, A(2,7), B(-4,-3): x2 - x1=-6, y2 - y1=-10. So (-6)^2=36, (-10)^2 = 100, sum 136, square root of 136 is about 11.66. BC: (6 - (-4))=10, (7 - (-3))=10, so 10^2+10^2=200, square root ~14.14. AC is 4. So 4+11.66+14.14=29.8. So perimeter is approximately 29.8.

Wait, but wait, maybe I messed up the points. Wait, A(2,7), C(6,7): horizontal line, length 4. B(-4,-3). So AB: from (2,7) to (-4,-3): left 6, down 10. So distance sqrt(36 + 100)=sqrt(136)≈11.66. BC: from (-4,-3) to (6,7): right 10, up 10, distance sqrt(100 + 100)=sqrt(200)≈14.14. So 4+11.66=15.66+14.14=29.8. So the perimeter is approximately 29.8.

Answer:

D. 29.8