How to find the height of of an acute / obtuse isosceles triangle - Geometry

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Question

An isosceles triangle has a base of $12\ cm$ and an area of $42\ cm^{2}$ . What must be the height of this triangle?

Answer

$A=\frac{1}{2}bh$ .

$6x=42$

$x=7$

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Question

An isosceles triangle has a perimeter of . If the base of the triangle is two less than two times the length of each leg, what is the height of the triangle?

Answer

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be .

Use the information given about the perimeter to solve for .

Plug this value in to find the length of the base.

$\text{Length of Base}=2(6)-2=10$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{6^2-5^2}=\sqrt{11}=3.32$

Make sure to round to places after the decimal.

Compare your answer with the correct one above

Question

An isosceles triangle has a perimeter of . If the base of the triangle is less than three times the length of a leg, what is the height of the triangle?

Answer

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be .

Use the information given about the perimeter to solve for .

Plug this value in to find the length of the base.

$\text{Length of Base}=3(10)-18=12$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{10^2-6^2}=\sqrt{64}=8$

Make sure to round to places after the decimal.

Compare your answer with the correct one above

Question

An isosceles triangle has a perimeter of . If the length of the base is one less than twice the length of a leg, what is the height of the triangle?

Answer

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be .

Use the information given about the perimeter to solve for .

Plug this value in to find the length of the base.

$\text{Length of Base}=2(5)-1=9$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{5^2-4.5^2}=\sqrt{4.75}=2.18$

Make sure to round to places after the decimal.

Compare your answer with the correct one above

Question

An isosceles triangle has a perimeter of . If the length of the base is fourteen less than three times the length of a leg, what is the height of the triangle?

Answer

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be .

Use the information given about the perimeter to solve for .

Plug this value in to find the length of the base.

$\text{Length of Base}=3(6)-14=4$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{6^2-2^2}=\sqrt{32}=5.66$

Make sure to round to places after the decimal.

Compare your answer with the correct one above

Question

An isosceles triangle has a perimeter of . If the length of the base is seven more than one-seventh of the length of a leg, what is the height of the triangle?

Answer

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be $\frac{1}{7}x+7$ .

Use the information given about the perimeter to solve for .

$x+x+\frac{1}{7}x+7=22$

$\frac{15}{7}x+7=22$

$\frac{15}{7}x=15$

Plug this value in to find the length of the base.

$\text{Length of Base}=\frac{1}{7}(7)+1=2$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{7^2-1^2}=\sqrt{33}=6.93$

Make sure to round to places after the decimal.

Compare your answer with the correct one above

Question

An isosceles triangle has a perimeter of . If the length of the base is two less than one-third of the length of a leg, what is the height of the triangle?

Answer

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be $\frac{1}{3}x-2$ .

Use the information given about the perimeter to solve for .

$x+x+\frac{1}{3}x-2=19$

$\frac{7}{3}x-2=19$

$\frac{7}{3}x=21$

Plug this value in to find the length of the base.

$\text{Length of Base}=\frac{1}{3}(9)-2=1$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{9^2-(\frac{1}{2})^2}=\sqrt{80.75}=8.99$

Make sure to round to places after the decimal.

Compare your answer with the correct one above

Question

An isosceles triangle has a perimeter of . If the length of the base is ten less than twice the length of a leg, what is the height of the triangle?

Answer

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be .

Use the information given about the perimeter to solve for .

Plug this value in to find the length of the base.

$\text{Length of Base}=2(12)-10=14$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{12^2-7^2}=\sqrt{95}=9.75$

Make sure to round to places after the decimal.

Compare your answer with the correct one above

Question

An isosceles triangle has a perimeter of . If the length of the base is four more than one-third of the length of a leg, what is the height of the triangle?

Answer

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be $\frac{1}{3}x+4$ .

Use the information given about the perimeter to solve for .

$x+x+\frac{1}{3}x+4=11$

$\frac{7}{3}x+4=11$

$\frac{7}{3}x=7$

Plug this value in to find the length of the base.

$\text{Length of Base}=\frac{1}{3}(3)+4=5$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{3^2-(\frac{5}{2})^2}=\sqrt{2.75}=1.66$

Make sure to round to places after the decimal.

Compare your answer with the correct one above

Question

The perimeter of an isosceles triangle is . If the length of the base is five more than one-fourth the length of a leg, what is the height of the triangle?

Answer

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be $\frac{1}{4}x+5$ .

Use the information given about the perimeter to solve for .

$x+x+\frac{1}{4}x+5=50$

$\frac{9}{4}x+5=50$

$\frac{9}{4}x=45$

Plug this value in to find the length of the base.

$\text{Length of Base}=\frac{1}{4}(20)+5=10$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{20^2-5^2}=\sqrt{375}=19.36$

Make sure to round to places after the decimal.

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Question

The perimeter of an isosceles triangle is . If the length of the base is ten more than one-eighth the length of a leg, what is the height of the triangle?

Answer

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be $\frac{1}{8}x+10$ .

Use the information given about the perimeter to solve for .

$x+x+\frac{1}{8}x+10=44$

$\frac{17}{8}x+10=44$

$\frac{17}{8}x=34$

Plug this value in to find the length of the base.

$\text{Length of Base}=\frac{1}{8}(16)+10=12$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{16^2-6^2}=\sqrt{220}=14.83$

Make sure to round to places after the decimal.

Compare your answer with the correct one above

Question

The perimeter of an isosceles triangle is . If the length of the base is eleven less than twice the length of a leg, what is the height of the triangle?

Answer

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be .

Use the information given about the perimeter to solve for .

Plug this value in to find the length of the base.

$\text{Length of Base}=2(7)-11=3$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{7^2-1.5^2}=\sqrt{46.75}=6.84$

Make sure to round to places after the decimal.

Compare your answer with the correct one above

Question

The perimeter of an isosceles triangle is . If the length of the base is fifteen more than one-third the length of a leg, what is the height of the triangle?

Answer

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be $\frac{1}{3}x+15$ .

Use the information given about the perimeter to solve for .

$x+x+\frac{1}{3}x+15=50$

$\frac{7}{3}x+15=50$

$\frac{7}{3}x=35$

Plug this value in to find the length of the base.

$\text{Length of Base}=\frac{1}{3}(15)+15=20$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{15^2-10^2}=\sqrt{125}=11.18$

Make sure to round to places after the decimal.

Compare your answer with the correct one above

Question

An isosceles triangle has a base of $12\ cm$ and an area of $42\ cm^{2}$ . What must be the height of this triangle?

Answer

$A=\frac{1}{2}bh$ .

$6x=42$

$x=7$

Compare your answer with the correct one above

Question

An isosceles triangle has a perimeter of . If the base of the triangle is two less than two times the length of each leg, what is the height of the triangle?

Answer

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be .

Use the information given about the perimeter to solve for .

Plug this value in to find the length of the base.

$\text{Length of Base}=2(6)-2=10$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{6^2-5^2}=\sqrt{11}=3.32$

Make sure to round to places after the decimal.

Compare your answer with the correct one above

Question

An isosceles triangle has a perimeter of . If the base of the triangle is less than three times the length of a leg, what is the height of the triangle?

Answer

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be .

Use the information given about the perimeter to solve for .

Plug this value in to find the length of the base.

$\text{Length of Base}=3(10)-18=12$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{10^2-6^2}=\sqrt{64}=8$

Make sure to round to places after the decimal.

Compare your answer with the correct one above

Question

An isosceles triangle has a perimeter of . If the length of the base is one less than twice the length of a leg, what is the height of the triangle?

Answer

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be .

Use the information given about the perimeter to solve for .

Plug this value in to find the length of the base.

$\text{Length of Base}=2(5)-1=9$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{5^2-4.5^2}=\sqrt{4.75}=2.18$

Make sure to round to places after the decimal.

Compare your answer with the correct one above

Question

An isosceles triangle has a perimeter of . If the length of the base is fourteen less than three times the length of a leg, what is the height of the triangle?

Answer

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be .

Use the information given about the perimeter to solve for .

Plug this value in to find the length of the base.

$\text{Length of Base}=3(6)-14=4$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{6^2-2^2}=\sqrt{32}=5.66$

Make sure to round to places after the decimal.

Compare your answer with the correct one above

Question

An isosceles triangle has a perimeter of . If the length of the base is seven more than one-seventh of the length of a leg, what is the height of the triangle?

Answer

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be $\frac{1}{7}x+7$ .

Use the information given about the perimeter to solve for .

$x+x+\frac{1}{7}x+7=22$

$\frac{15}{7}x+7=22$

$\frac{15}{7}x=15$

Plug this value in to find the length of the base.

$\text{Length of Base}=\frac{1}{7}(7)+1=2$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{7^2-1^2}=\sqrt{33}=6.93$

Make sure to round to places after the decimal.

Compare your answer with the correct one above

Question

An isosceles triangle has a perimeter of . If the length of the base is two less than one-third of the length of a leg, what is the height of the triangle?

Answer

First, find the lengths of the triangle.

Let be the length of each leg. Then, the length of the base must be $\frac{1}{3}x-2$ .

Use the information given about the perimeter to solve for .

$x+x+\frac{1}{3}x-2=19$

$\frac{7}{3}x-2=19$

$\frac{7}{3}x=21$

Plug this value in to find the length of the base.

$\text{Length of Base}=\frac{1}{3}(9)-2=1$

Now, recall that the height of an isosceles triangle can split the entire triangle into two congruent right triangle as shown by the figure below.

Thus, we can use the Pythagorean Theorem to find the length of the height.

$\text{Height}=\sqrt{\text{leg}^2-\text{half of base}^2}$

Plug in the given values to find the height of the triangle.

$\text{Height}=\sqrt{9^2-(\frac{1}{2})^2}=\sqrt{80.75}=8.99$

Make sure to round to places after the decimal.

Compare your answer with the correct one above

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