Question

The vertices of ∆ABC are A(2, 8), B(16, 2), and C(6, 2). what is the perimeter and area in square units

Answer 1

The perimeter of ∆ABC with vertices at A(2, 8), B(16, 2), and C(6, 2) is approximately 32.44 units, and the area is approximately 866.11 square units, calculated using the distance formula and Heron's formula.

Step-by-step explanation:

To calculate the perimeter and area of ∆ABC with vertices at A(2, 8), B(16, 2), and C(6, 2), we use the distance formula to find the lengths of the sides of the triangle and then apply those values to find the perimeter and area.

The distance formula is √((x2-x1)^2 + (y2-y1)^2), which gives us the lengths of AB, BC, and AC:

• AB: √((16-2)^2 + (2-8)^2) = √(196 + 36) = √232
• BC: √((6-16)^2 + (2-2)^2) = √(100 + 0) = 10
• AC: √((6-2)^2 + (2-8)^2) = √(16 + 36) = √52

The perimeter (P) is AB + BC + AC, and to find the area (A) we could use Heron's formula, which requires the semi-perimeter (s = P/2).

Calculating the perimeter:

P = √232 + 10 + √52 = (√232) + 10 + (√52) ≈ 15.23 + 10 + 7.21 ≈ 32.44 units

For the area, we first calculate the semi-perimeter:

s = P/2 ≈ 32.44/2 ≈ 16.22 units

Then apply Heron's formula:

A = √(s(s-AB)(s-BC)(s-AC)) = √(16.22(16.22-√232)(16.22-10)(16.22-√52))

After calculations:

A ≈ √(16.22(1)(6.22)(9)) ≈ √866.11 square units

Therefore, the perimeter of ∆ABC is approximately 32.44 units and the area is approximately 866.11 square units.

Answer 2

Best is to draw a sketch of the three points.
Next step is to find the distances BC, CD, DB.
The perimeter is the sum of the three distances.

The distances are found using the distance formula:
D=sqrt((y2-y1)^2+(x2-x1)^2)
order of (x1,y1), (x2,y2) is not important.

Given A(2,8),B(16,2),C(6,2)
we calculate
AB=sqrt((16-2)^2+(2-8)^2)=sqrt(14^2+6^2)=sqrt(232);
BC=sqrt((6-16)^2+(2-2)^2)=sqrt(10^2+0)=10
CA=sqrt((2-6)^2+(8-2)^2)=sqrt(4^2+6^2)=sqrt(16+36)=sqrt(52)

Perimeter=AB+BC+CA=32.443 units

For the area, we note that BC is horizontal (parallel to the x-axis), so
area = (1/2)bast * height
=(1/2)10*(ya-yb)
=(1/2)10*(8-2)
=(1/2)10*6
=30 unit^2