Height Of Isosceles Triangle Formula

The height of an isosceles triangle is the perpendicular altitude from the base of the triangle to the opposite vertex. The elevation of a triangle is one of its important dimensions considering it allows us to calculate the area of the triangle.

To detect the length of the height of an isosceles triangle, nosotros accept to apply the Pythagoras theorem to derive a formula.

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formula for the height of an isosceles triangle

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Learning about the summit of an isosceles triangle with examples.

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GEOMETRY

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Learning almost the height of an isosceles triangle with examples.

Come across examples

Formula for the height of an isosceles triangle

The acme of an isosceles triangle is calculated using the length of its base and the length of one of the congruent sides. We tin calculate the height using the following formula:

$latex h= \sqrt{{{a}^ii}- \frac{{{b}^2}}{4}}$

wherea is the length of the coinciding sides of the triangle andb is the length of the base of operations of the triangle.

Derivation of the elevation formula

To derive this formula, we can consider the following isosceles triangle:

diagram of height of an isosceles triangle

By drawing a line representing the height, we can run into that nosotros divide the isosceles triangle into two congruent right triangles. We can use one of the obtained triangles and apply the Pythagorean theorem to calculate the height.

Recollect that the Pythagorean theorem does not say that the square of the hypotenuse is equal to the sum of the squares of the legs. Therefore, we accept:

$latex {{a}^ii}={{h}^2}+{{( \frac{b}{ii})}^two}$

$latex {{a}^ii}={{h}^ii}+ \frac{{{b}^two}}{iv}$

$latex {{h}^two}={{a}^2}- \frac{{{b}^2}}{four}$

$latex h= \sqrt{{{a}^2}- \frac{{{b}^2}}{4}}$

We accept obtained an expression for the superlative.

Height of an isosceles triangle – Examples with answers

The following examples use the seen formula to discover the height of isosceles triangles. Endeavour to solve the exercises yourself before looking at the solution.

EXAMPLE 1

What is the height of an isosceles triangle that has a base of 8 thousand and congruent sides of length 6 1000?

Solution

From the question, we have the post-obit data:

Base, $latex b=eight$ yard
Sides, $latex a=half dozen$ one thousand

Therefore, we use the peak formula with these values:

$latex h= \sqrt{{{a}^ii}- \frac{{{b}^ii}}{4}}$

$latex h= \sqrt{{{half dozen}^2}- \frac{{{8}^two}}{4}}$

$latex h= \sqrt{36- \frac{64}{four}}$

$latex h= \sqrt{36-16}$

$latex h= \sqrt{20}$

$latex h=4.47$

The pinnacle of the triangle is 4.47 m.

EXAMPLE 2

An isosceles triangle has a base of ten m and congruent sides of length 12 g. What is the length of its acme?

Solution

We can identify the following data:

Base, $latex b=10$ g
Sides, $latex a=12$ m

Substituting these values in the formula, we accept:

$latex h= \sqrt{{{a}^2}- \frac{{{b}^2}}{4}}$

$latex h= \sqrt{{{12}^2}- \frac{{{10}^2}}{iv}}$

$latex h= \sqrt{144- \frac{100}{4}}$

$latex h= \sqrt{144-25}$

$latex h= \sqrt{119}$

$latex h=10.91$

The acme of the triangle is 10.91 m.

Instance iii

An isosceles triangle has a base of operations of length 8 m and coinciding sides of length 9 k. What is the length of the height?

Solution

From the question, we take the post-obit values:

Base of operations, $latex b=8$ m
Sides, $latex a=ix$ m

Substituting these values in the tiptop formula, we have:

$latex h= \sqrt{{{a}^2}- \frac{{{b}^two}}{4}}$

$latex h= \sqrt{{{9}^2}- \frac{{{8}^two}}{4}}$

$latex h= \sqrt{81- \frac{64}{four}}$

$latex h= \sqrt{81-16}$

$latex h= \sqrt{65}$

$latex h=eight.06$

The pinnacle of the triangle is 8.06 m.

Instance 4

What is the height of a triangle that has a base of length 14 m and congruent sides of length xi g?

Solution

We take the following data:

Base, $latex b=14$ m
Sides, $latex a=11$ m

Therefore, we utilize the height formula with these values:

$latex h= \sqrt{{{a}^2}- \frac{{{b}^2}}{4}}$

$latex h= \sqrt{{{xi}^2}- \frac{{{14}^2}}{4}}$

$latex h= \sqrt{121- \frac{196}{4}}$

$latex h= \sqrt{121-49}$

$latex h= \sqrt{72}$

$latex h=8.49$

The height of the triangle is 8.49 chiliad.

Height of an isosceles triangle – Practise bug

Use the formula for the height of isosceles triangles to solve the following problems. If y'all demand help, you can look at the solved examples higher up.

What is the elevation of an isosceles triangle with a base of 6m and sides of length 5m?

Choose an answer

$latex h=ii.8m$

$latex h=3.8m$

$latex h=4m$

$latex h=4.5m$

What is the height of an isosceles triangle with a base of 8m and sides of length 10m?

Choose an answer

$latex h=8.27m$

$latex h=8.97m$

$latex h=9.17m$

$latex h=10.37m$

An isosceles triangle has a base of length 15 m and sides of length 24 chiliad. What is its peak?

Choose an answer

$latex h=twenty.6m$

$latex h=21.2m$

$latex h=22.1m$

$latex h=22.8m$

Run into too

Interested in learning more nearly isosceles triangles? Take a look at these pages:

Area of an Isosceles Triangle – Formulas and Examples
Perimeter of an Isosceles Triangle – Formulas and Examples
What are the characteristics of isosceles triangles?
Isosceles Acute Triangle – Characteristics and Examples
Isosceles Obtuse Triangle – Characteristics and Examples

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