A triangle has sides with lengths of 5, 7, and 3. What is the radius of the triangles inscribed circle?

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A triangle has sides with lengths of 5, 7, and 3. What is the radius of the triangles inscribed circle?

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Answer 1 · 2016-10-27T11:04:43

Radius $= \frac{\sqrt{3}}{2}$ $=sqrt(3)/2$

Explanation:

[1] Step 1: Find the Area of the Triangle
Use Heron's formula: $A r e a_{△} = \sqrt{s (s - a) (s - b) (s - c)}$ $Area_triangle=sqrt(s(s-a)(s-b)(s-c))$
where $a, b, c$ $a, b, c$ are the lengths of the sides and $s$ $s$ is the semi-perimeter $\frac{a + b + c}{2}$ $(a+b+c)/2$

$XXX s = \frac{5 + 7 + 3}{2} = \frac{15}{2}$ $color(white)("XXX")s=(5+7+3)/2=15/2$

$XXX A r e a_{△} = \sqrt{\frac{15}{2} \cdot \frac{5}{2} \cdot \frac{1}{2} \cdot \frac{9}{2}}$ $color(white)("XXX")Area_triangle = sqrt(15/2 * 5/2 * 1/2 * 9/2)$

$XXXXXXX = \frac{15}{4} \sqrt{3}$ $color(white)("XXXXXXX")=15/4sqrt(3)$

[2] Step 2: Find the Radius of Inscribed Circle
Use teh forumla: ${Radius}_{\circ} = \frac{{Area}_{△}}{s}$ $"Radius"_circ = "Area"_triangle/s$
$XXXXXXXXX$ $color(white)("XXXXXXXXX")$ If you are unfamiliar with this formula:
$XXXXXXXXX$ $color(white)("XXXXXXXXX")$ ask to have it explained as a separate question.

$XXX r_{\circ} = \frac{\frac{15}{4} \sqrt{3}}{\frac{15}{2}} = \frac{\sqrt{3}}{2}$ $color(white)("XXX")r_circ = (15/4sqrt(3))/(15/2)=sqrt(3)/2$

~~~~~~ Oops! The next step wasn't asked for but I did it anyway ~~~~~
[3] Calculate the Area of the Circle
Use the formula: ${Area}_{\circ} = π r^{2}$ $"Area"_circ = pir^2$

$XXX {Area}_{\circ} = π {(\frac{\sqrt{3}}{2})}^{2} = \frac{3}{4} π$ $color(white)("XXX")"Area"_circ = pi(sqrt(3)/2)^2=3/4pi$

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A triangle has sides with lengths of 5, 7, and 3. What is the radius of the triangles inscribed circle?

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