Solved on Jan 20, 2024

Find if $\triangle ABC$ with slopes $\frac{3}{4}$ and $-\frac{4}{3}$ is isosceles, right-angled, parallel, or equilateral.

STEP 1

Assumptions
1. We are given a triangle $\triangle ABC$ .
2. The slope of line segment $AB$ is $\frac{3}{4}$ .
3. The slope of line segment $BC$ is $-\frac{4}{3}$ .
4. We are to determine the nature of the triangle based on the given slopes.

STEP 2

Recall the relationship between the slopes of two perpendicular lines. If two lines are perpendicular, the product of their slopes is $-1$ .

STEP 3

Calculate the product of the given slopes.
$Product\, of\, slopes = \left(\frac{3}{4}\right) \times \left(-\frac{4}{3}\right)$

STEP 4

Simplify the product.
$Product\, of\, slopes = \left(\frac{3}{4}\right) \times \left(-\frac{4}{3}\right) = -1$

STEP 5

Since the product of the slopes is $-1$ , we can conclude that the line segments $AB$ and $BC$ are perpendicular to each other.

STEP 6

If two sides of a triangle are perpendicular to each other, then the triangle is a right-angled triangle.

STEP 7

Based on Step 6, we can conclude that $\triangle ABC$ is a right-angled triangle.
Therefore, the correct answer is (b) $\triangle ABC$ is right-angled.

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Find if $\triangle ABC$ with slopes $\frac{3}{4}$ and $-\frac{4}{3}$ is isosceles, right-angled, parallel, or equilateral.

STEP 1

Assumptions
1. We are given a triangle $\triangle ABC$ .
2. The slope of line segment $AB$ is $\frac{3}{4}$ .
3. The slope of line segment $BC$ is $-\frac{4}{3}$ .
4. We are to determine the nature of the triangle based on the given slopes.

STEP 2

Recall the relationship between the slopes of two perpendicular lines. If two lines are perpendicular, the product of their slopes is $-1$ .

STEP 3

Calculate the product of the given slopes.
$Product\, of\, slopes = \left(\frac{3}{4}\right) \times \left(-\frac{4}{3}\right)$

STEP 4

Simplify the product.
$Product\, of\, slopes = \left(\frac{3}{4}\right) \times \left(-\frac{4}{3}\right) = -1$

STEP 5

Since the product of the slopes is $-1$ , we can conclude that the line segments $AB$ and $BC$ are perpendicular to each other.

STEP 6

If two sides of a triangle are perpendicular to each other, then the triangle is a right-angled triangle.

STEP 7

Based on Step 6, we can conclude that $\triangle ABC$ is a right-angled triangle.
Therefore, the correct answer is (b) $\triangle ABC$ is right-angled.

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Find if △ABC\triangle ABC△ABC with slopes 34\frac{3}{4}43​ and −43-\frac{4}{3}−34​ is isosceles, right-angled, parallel, or equilateral.

STEP 1

Assumptions1. We are given a triangle △ABC\triangle ABC△ABC.2. The slope of line segment ABABAB is 34\frac{3}{4}43​.3. The slope of line segment BCBCBC is −43-\frac{4}{3}−34​.4. We are to determine the nature of the triangle based on the given slopes.

STEP 2

Recall the relationship between the slopes of two perpendicular lines. If two lines are perpendicular, the product of their slopes is −1-1−1.

STEP 3

Calculate the product of the given slopes.Product of slopes=(34)×(−43)Product\, of\, slopes = \left(\frac{3}{4}\right) \times \left(-\frac{4}{3}\right)Productofslopes=(43​)×(−34​)

STEP 4

Simplify the product.Product of slopes=(34)×(−43)=−1Product\, of\, slopes = \left(\frac{3}{4}\right) \times \left(-\frac{4}{3}\right) = -1Productofslopes=(43​)×(−34​)=−1

STEP 5

Since the product of the slopes is −1-1−1, we can conclude that the line segments ABABAB and BCBCBC are perpendicular to each other.

STEP 6

If two sides of a triangle are perpendicular to each other, then the triangle is a right-angled triangle.

STEP 7

Based on Step 6, we can conclude that △ABC\triangle ABC△ABC is a right-angled triangle.Therefore, the correct answer is (b) △ABC\triangle ABC△ABC is right-angled.

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Find if $\triangle ABC$ with slopes $\frac{3}{4}$ and $-\frac{4}{3}$ is isosceles, right-angled, parallel, or equilateral.

Assumptions
1. We are given a triangle $\triangle ABC$ .
2. The slope of line segment $AB$ is $\frac{3}{4}$ .
3. The slope of line segment $BC$ is $-\frac{4}{3}$ .
4. We are to determine the nature of the triangle based on the given slopes.

Recall the relationship between the slopes of two perpendicular lines. If two lines are perpendicular, the product of their slopes is $-1$ .

Calculate the product of the given slopes.
$Product\, of\, slopes = \left(\frac{3}{4}\right) \times \left(-\frac{4}{3}\right)$

Simplify the product.
$Product\, of\, slopes = \left(\frac{3}{4}\right) \times \left(-\frac{4}{3}\right) = -1$

Since the product of the slopes is $-1$ , we can conclude that the line segments $AB$ and $BC$ are perpendicular to each other.

Based on Step 6, we can conclude that $\triangle ABC$ is a right-angled triangle.
Therefore, the correct answer is (b) $\triangle ABC$ is right-angled.