A plot of the points A(-2, 11), B(5, 7), C(1, 0) is given by the option;
- The triangle is a right triangle given that (AC)² = (AB)² + (BC)²
- The area of the ∆ABC is 32.5 square units.
- Sum of the squares of the lengths of the legs of the triangle = <u>130</u>
- Square of the length of the hypotenuse of the triangle = <u>130</u>
<h3>How can ∆ABC be proven to be a right triangle from its dimensions?</h3>
The coordinates of the vertices of the triangle are;
A = (-2, 11), B = (5, 7), C = (1, 0)
Therefore, on the coordinate plane, we have;
The highest and leftmost point of the triangle is the vertex, <em>A</em>
The second highest and rightmost point of the triangle is the vertex, <em>B</em>
The<em> </em>vertex of the triangle that is midway between <em>A </em>and <em>B </em>and the lowest vertex of the triangle is the vertex <em>C</em>
- The<em> </em>correct<em> </em>option that shows the points ABC and triangle ABC is the option <em>D</em>.
The lengths of the sides of the triangle are;
AB = √((-2 - 5)² + (11 - 7)²) = √(65)
BC = √((5 - 1)² + (7 - 0)²) = √(65)
AC = √((-2 - 1)² + (11 - 0)²) = √(130)
Therefore;
(AC)² = 130 = (AB)² + (BC)² = 65 + 65
Which gives;
Therefore;
- ∆ABC is a right triangle, from the definition of a right triangle.
The legs of ∆ABC are AB and BC
AC is the hypotenuse of ∆ABC
The area of ∆ABC is therefore;
Area = (1/2) × AB × BC
Which gives;
Area of ∆ABC = (1/2) × √(65) × √(65)
√(65) × √(65) = 65
- Area of ∆ABC = (1/2) × 65 = 32.5
(AB)² + (BC)² = 65 + 65 = 130
Therefore;
- Sum of the squares of the lengths of the legs of the triangle = 130
(AC)² = 130
- Square of the length of the hypotenuse of the triangle = 130
Learn more about right triangles here:
brainly.com/question/2284306
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