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Chapter 14: Problem 1307

A triangle with vertices \((4,0),(-1,-1),(3,5)\) is (a) isosceles and right angled (b) isosceles but not right angled (c) right angled but not isosceles (d) neither right angled nor isosceles

Short Answer

Expert verified
The triangle with vertices \((4,0)\), \((-1,-1)\), and \((3,5)\) has side lengths \(AB = AC = \sqrt{26}\) and \(BC = \sqrt{52}\). Since two sides have equal lengths, the triangle is isosceles. Additionally, the triangle is right-angled, as the square of the longest side equals the sum of the squares of the other two sides: \(BC^2 = AB^2 + AC^2\). Therefore, the correct answer is option (a): the triangle is isosceles and right-angled.

Step by step solution

01

Calculate the side lengths

We'll start by finding the length of each side of the triangle using the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). To find the length of side AB between points \((4,0)\) and \((-1,-1)\), we plug in the points into the distance formula: \(AB = \sqrt{(-1 - 4)^2 + (-1 - 0)^2} = \sqrt{25 + 1} = \sqrt{26}\) Similarly, for side BC (between points \((-1,-1)\) and \((3,5)\)): \(BC = \sqrt{(3 - (-1))^2 + (5 - (-1))^2} = \sqrt{16 + 36} = \sqrt{52}\) And for side AC (between points \((4,0)\) and \((3,5)\)): \(AC = \sqrt{(3 - 4)^2 + (5 - 0)^2} = \sqrt{1 + 25} = \sqrt{26}\) So, we find that the side lengths are: \(AB = \sqrt{26}\) \(BC = \sqrt{52}\) \(AC = \sqrt{26}\)
02

Determine if the triangle is isosceles

The triangle is isosceles if at least two of its sides have equal lengths. From the side lengths calculated in step 1, we see that sides AB and AC have equal lengths: \(AB = AC = \sqrt{26}\) Since two sides have equal length, the triangle is isosceles.
03

Determine if the triangle is right-angled

To determine if the triangle is right-angled, we check if the square of the longest side is equal to the sum of the squares of the other two sides (Pythagorean theorem): \(BC^2 = AB^2 + AC^2\) Substitute the side lengths: \((\sqrt{52})^2 = (\sqrt{26})^2 + (\sqrt{26})^2\) \(52 = 26 + 26\) This equality holds true, so the triangle is right-angled. #Conclusion#Now that we have determined that the triangle is both isosceles and right-angled, we can confidently choose option (a) as the correct answer. The triangle is isosceles and right-angled.

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