Q&A for How to Construct an Isosceles Triangle

If you know the length of the hypotenuse, you can find the length of the other two sides of the triangle using the Pythagorean theorem (a^2 + b^2 = c^2). However, since this is an isosceles triangle, the two sides will be the same length, so you will simplify the Pythagorean formula to x^2 + x^2 = c^2, or 2x^2 = c^2. For example, if the hypotenuse is 12 cm, the formula will be 2x^2 = 12^2: 2x^2 = 12^2 2x^2 = 144 2x^2/2 = 144/2 x^2 = 72 sqrt*x^2 = sqrt*72 x = 8.48. Since every triangle has 180 degrees, if it is a right triangle, the angle measurements are 90-45-45. So the triangle will have a hypotenuse of 12, two side lengths of about 8.5 cm, and two 45 degree angles.

You don't have enough information to do that.

Both base angles are 45°. Therefore the third angle is 90°. Drop an altitude from the 90° angle to the base. The altitude bisects the 90° angle. It also bisects the base and is perpendicular to it. The altitude forms two smaller isosceles right triangles, each of which has two 45° angles and two sides with lengths of 30 (half the base). Thus, each 45° angle in each smaller right triangle has an opposite side and an adjacent side of length 30 and a hypotenuse of x (the length you're trying to find). The sine (and cosine) of each 45° angle is 0.707. Therefore, 0.707 = 30 / x. x = 30 / 0.707 = 42.4. That's the length of each of the two equal sides of the big triangle.

Draw straight line segment AB 10 cm in length. Using a compass, from point A draw an arc of 5 cm radius. From point B draw an arc of 8 cm radius. The intersection of the two arcs is point C. Draw straight lines from A to C and from B to C. You have your triangle.

Draw a line segment 10 cm in length, and label it AC. Then choose any convenient length for AB and BC, and proceed as shown in Method 1.

Use a protractor to draw a 75° angle with vertex point A. Mark an equal distance from the vertex along both rays of the angle (at points B and C). Draw a line connecting B and C. That forms an isosceles triangle.

Draw the base. From each end of the base mark off an arc with a radius equal to the length of the other side(s). The intersection of the arcs is the third vertex of the triangle. Connect the vertices. That's your triangle.

You don't have enough information to construct the triangle. For one thing, you don't know whether the hypotenuse is larger or smaller than the other sides.

An isosceles triangle is one containing two (and only two) equal sides. The angles opposite the equal sides are also equal. The altitude drawn to the base (the non-equal side) of an isosceles triangle bisects the angle from which it's drawn. The altitude also bisects and is perpendicular to the base.

Draw a straight line 100 mm in length. Consider it the base of the triangle. At any point on the base construct a perpendicular line 70 mm in length. Draw straight lines from the far end of the 70 mm line to each end of the 100 mm line. That forms a triangle. If it's to be an isosceles triangle, construct the perpendicular line at the midpoint of the base.

Call the triangle ABC with base BC. Draw BC of length 3 cm. Using a protractor, from B draw a line 45° from BC. On the same side of BC, from C draw another line 45° from BC. The intersection of these two lines is point A.

See Construct a Bisector of a Given Angle .

See Method 2 above.

This can't be done. An isosceles triangle with a vertex angle of 60° would also have base angles of 60° each, meaning that the triangle is equiangular and thus equilateral. That means it couldn't have a base length different from its side lengths.

You would need more information before you could perform that construction.

You're not given enough information to perform that construction.

First draw median AD of length 4.2 cm. Draw a line of undetermined length perpendicular to AD at D. That line will become base BC. From A draw an arc of radius 4.9 cm intersecting BC at B and C. You now have points A, B and C, so you can draw the triangle.