Question

Find the length of each side of the triangle determined by the three points P1,P2,P 1,P 2, and P3. P3 State whether the triangle is an isosceles triangle, a right triangle, neither of these, or both. (An isosceles triangle is one in which at least two of the sides are of equal length.) P1=(−1,4);P2=(6,2);P3=(4,−5)P 1=(−1,4);P 2=(6,2);P 3 =(4,−5).

Answer 1

IT IS BOTH

Explanation:

Given the coordinate of the triangle, P1P2P3, we are to classify the triangle. To do that we need to know the length of each side of the triangle

P1P2, P2P3 and P1P2 using the formula for calculating distance between two coordinates

D = √(x2-x1)²+(y2-y1)²

For P1P2;

P1=(−1,4) P2=(6,2)

P1P2 = √(6-(-1))²+(2-4)²

P1P2 = √7²+2²

P1P2 = √53

For P1P3;

P1=(−1,4) P3=(4,-5)

P1P3 = √(4-(-1))²+(-5-4)²

P1P3 = √5²+9²

P1P3 = √25+81

P1P3 = √106

For P2P3;

P2=(6,2) P3 = (4,-5)

P2P3= √(4-6)²+(-5-2)²

P2P3 = √2²+7²

P2P3 = √53

Since two sides of the triangles are the same, this means that it is an isosceles triangle.

Let's check whether it is right angled by applying Pythagoras theorem

(√53)²+(√53)² = x²

x is the longest side (hypotenuse)

53+53 = x²

106 = x²

x = √106

Since the square of the hypotenuse is equal to the sum of the sum of the square of other two sides, then it is also a RIGHT ANGLED TRIANGLE