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A piece of steel is in the shape of a triangle. One side of the triangle is 8 inches long. The lengths of the other two sides are equal and each is less than 8 inches long.

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A. What type of triangle must describe the shape of the piece of steel?

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Research triangle types

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A triangle has three sides and three angles

The three angles always add to 180°

Equilateral, Isosceles and Scalene

There are three special names given to triangles that tell how many sides (or angles) are equal.

There can be 3, 2 or no equal sides/angles:

Equilateral Triangle

Three equal sides
Three equal angles, always 60°

Isosceles Triangle

Two equal sides
Two equal angles

Scalene Triangle

No equal sides
No equal angles

How to remember? Alphabetically they go 3, 2, none:

Equilateral: "equal"-lateral (lateral means side) so they have all equal sides

Isosceles: means "equal legs", and we have two legs, right? Also iSOSceles has two equal "Sides" joined by an "Odd" side.

Scalene: means "uneven" or "odd", so no equal sides.

What Type of Angle?

Triangles can also have names that tell you what type of angle is inside:

Acute Triangle

All angles are less than 90°

Right Triangle

Has a right angle (90°)

Obtuse Triangle

Has an angle more than 90°

Combining the Names
Sometimes a triangle will have two names, for example:

Right Isosceles Triangle

Has a right angle (90°), and also two equal angles

Can you guess what the equal angles are?

Play With It ...

Try dragging the points around and make different triangles:

geometry/images/triangle.js?mode=type

You might also like to play with the Interactive Triangle.

Angles

The three interior angles always add to 180°

geometry/images/triangle.js?mode=angles

Perimeter

The perimeter is the distance around the edge of the triangle: just add up the three sides:

geometry/images/triangle.js?mode=perim

Area

The area is half of the base times height.

"b" is the distance along the base

"h" is the height (measured at right angles to the base)

Area = ½ × b × h

The formula works for all triangles.

Note: a simpler way of writing the formula is bh/2

Example: What is the area of this triangle?
(Note: 12 is the height, not the length of the left-hand side)
Height = h = 12

Base = b = 20

Area = ½ × b × h = ½ × 20 × 12 = 120

The base can be any side, Just be sure the "height" is measured at right angles to the "base":

geometry/images/triangle.js?mode=area

(Note: You can also calculate the area from the lengths of all three sides using Heron's Formula.)

Why is the Area "Half of bh"?

Imagine you "doubled" the triangle (flip it around one of the upper edges) to make a square-like shape (a parallelogram) which can be changed to a simple rectangle:

THEN the whole area is bh, which is for both triangles, so just one is ½ × bh.

6702, 6708,720, 3134, 5032,627,723, 3132, 3133, 7502

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Triangles can be either isosceles, equilateral or scalene.

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Isosceles triangles have 2 sides and 2 angles that are the same.

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Can we use the Pythagorean Theorem to solve this?

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This has to be isosceles because two sides are the same. The question says two sides are equal and the other is 8 inches. The equal sides are less than 8 inche.

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B. What are possible values for each of the three angle measures, in degrees, of the vertices of the triangle?

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Create triangles
See which combinations work and which do not

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Another piece of steel is also in the shape of a triangle. One side of the triangle is 6 inches long. The lengths of the other two sides are equal and each is less than 6 inches long. The lengths of the unknown sides of the triangle are whole numbers of inches.

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C. What are all the possible lengths, in inches, of the unknown sides of the triangle? Explain the process you used to find all the possible lengths.