Math / Geometry

Question6. Classify the triangle with the following side lengths: $20,21,29$

Studdy Solution

STEP 1

1. We are given a triangle with side lengths $20$ , $21$ , and $29$ .
2. We need to classify the triangle as either scalene, isosceles, or equilateral.
3. We also need to determine if it is a right triangle.

STEP 2

1. Check if the triangle is valid using the triangle inequality theorem.
2. Classify the triangle based on its side lengths.
3. Determine if the triangle is a right triangle using the Pythagorean theorem.

STEP 3

Check the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side. Verify for all combinations:
$20 + 21 > 29$ $20 + 29 > 21$ $21 + 29 > 20$
Calculate each:
$41 > 29 \quad \text{(True)}$ $49 > 21 \quad \text{(True)}$ $50 > 20 \quad \text{(True)}$
Since all conditions are satisfied, the triangle is valid.

STEP 4

Classify the triangle based on its side lengths. A triangle is: - Equilateral if all three sides are equal. - Isosceles if exactly two sides are equal. - Scalene if all three sides are different.
Check the side lengths $20$ , $21$ , and $29$ :
All three sides are different, so the triangle is scalene.

STEP 5

Determine if the triangle is a right triangle using the Pythagorean theorem, which states that for a right triangle with sides $a$ , $b$ , and hypotenuse $c$ , $a^2 + b^2 = c^2$ .
Assume $29$ is the hypotenuse ( $c$ ) and check:
$20^2 + 21^2 = 29^2$
Calculate each:
$20^2 = 400$ $21^2 = 441$ $29^2 = 841$
Check the sum:
$400 + 441 = 841$
Since $841 = 841$ , the triangle is a right triangle.
The triangle with side lengths $20$ , $21$ , and $29$ is a scalene right triangle.

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