Question

plot each point and form the right triangle abc. verify that the triangle is a right triangle. find its area.
a = (-1,3); b = (6,0); c = (4,5)

choose the correct graph below that shows points a, b, c, and triangle abc.
a.
graph a
b.
graph b
c.
graph c
d.
graph d

to show that the triangle is a right triangle, show that the sum of the squares of the lengths of two of the sides (the legs) equals the square of the length of the third side (the hypotenuse).
the sum of the squares of the lengths of the legs of the triangle is
the square of the length of the hypotenuse of the triangle is

Explanation:

Step1: Calculate lengths of sides

Use distance formula $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$, square results:

• $AB^2=(6-(-1))^2+(0-3)^2=7^2+(-3)^2=49+9=58$
• $BC^2=(4-6)^2+(5-0)^2=(-2)^2+5^2=4+25=29$
• $AC^2=(4-(-1))^2+(5-3)^2=5^2+2^2=25+4=29$

Step2: Identify right angle check

Check Pythagorean theorem: $BC^2+AC^2=29+29=58=AB^2$

Step3: Sum of squares of legs

Sum of squares of the two shorter sides (legs):
$29+29=58$

Step4: Square of hypotenuse

Square of the longest side (hypotenuse $AB$):
$58$

Step5: Calculate area

Area = $\frac{1}{2} \times$ leg1 $\times$ leg2. Legs are $\sqrt{29}$ each:
$\text{Area}=\frac{1}{2} \times \sqrt{29} \times \sqrt{29}=\frac{1}{2} \times 29=14.5$

Answer:

Correct graph: D.
The sum of the squares of the lengths of the legs of the triangle is $58$.
The square of the length of the hypotenuse of the triangle is $58$.
Area of triangle ABC: $\frac{29}{2}$ or $14.5$