Question

Show directly, without the use of Ptolemy's theorem, that in an isosceles trapezoid, the square on a diagonal is equal to the sum of the product of the two parallel sides plus the square on one of the other sides.

Short answer

Question: In an isosceles trapezoid, show that the square on a diagonal is equal to the sum of the product of the two parallel sides plus the square on one of the other sides. Answer: To show this, we set up relationships between the sides and diagonal of the isosceles trapezoid using the Pythagorean theorem. After algebraic manipulation, we found that in an isosceles trapezoid, the square on a diagonal (d^2) is equal to the sum of the product of the two parallel sides (a^2 and b^2) plus the square on one of the other sides (c^2). Thus, d^2 = a^2 - 2ab + b^2 + c^2.

Step-by-step solution

Label the isosceles trapezoid, sides, and diagonal.

Let's call the isosceles trapezoid ABCD, where AB // CD, AB > CD, and AD = BC (due to the trapezoid being isosceles). Let E be the point where the diagonal AC intersects the other diagonal BD. Let AB = a, CD = b, AD = BC = c, and AC = d.

Identify triangles and relationships in the trapezoid.

We can see that triangle AEC and triangle DEC are right triangles with hypotenuse AC. This is because E is where the altitude from vertices A and C meet the opposite side, creating two right-angled triangles. Additionally, note that triangle ABE and triangle DCE are congruent due to AD = BC and angle A = angle C (due to opposite angles of the parallel sides).

Apply the Pythagorean theorem to triangles AEC and DEC.

The Pythagorean theorem states that the sum of squares on the legs of a right triangle is equal to the square on the hypotenuse. Thus, we have: For triangle AEC: \([AE^2 + (c - CE)^2 = d^2] \ (1)\) For triangle DEC: \([DE^2 + CE^2 = d^2] \ (2)\)

Apply the Pythagorean theorem to triangle ABE.

Let's call the length of AE and DE, x and y respectively. Then for triangle ABE, we have: \( x^2 + (a - b)^2 = c^2 \ (3)\) Similarly, triangle ABE is congruent to triangle DCE, we can write: \( y^2 + (a - b)^2 = c^2 \ (4)\)

Add equations (1) and (2).

By adding equations (1) and (2), we get: \([(x^2 + (c - CE)^2) + (y^2 + CE^2)] = 2d^2\)

Use equations (3) and (4) to relate back to the trapezoid sides.

Since equation (3) states that \(x^2 + (a - b)^2 = c^2\) and equation (4) states \(y^2 + (a - b)^2 = c^2\), we can substitute this back into the equation from Step 5: \([(c^2 - (a - b)^2) + (c^2 - (a - b)^2)] = 2d^2\)

Simplify the equation from Step 6.

After simplifying the equation from Step 6, we get: \([2c^2 - 2(a - b)^2] = 2d^2\) Divide both sides by 2: \(c^2 - (a - b)^2 = d^2\)

Expanding the equation and getting the desired result.

Now, we expand the equation from Step 7: \(c^2 - [a^2 - 2ab + b^2] = d^2\) \(a^2 - 2ab + b^2 + c^2 = d^2\) Hence, in an isosceles trapezoid, the square on a diagonal is equal to the sum of the product of the two parallel sides plus the square on one of the other sides.

Key concepts

Ptolemy's Theorem

Ptolemy's theorem is a fascinating and useful result in Euclidean geometry that ties together the lengths of sides and diagonals in a cyclic quadrilateral (a four-sided figure where all vertices lie on a single circle). Specifically, it states that the sum of the products of the opposite sides equals the product of the diagonals.

Although Ptolemy's theorem is not directly applied in the given problem, understanding it can deepen one's insight into the properties of quadrilaterals. While our focus here is on isosceles trapezoids, which are not cyclic, Ptolemy's theorem can provide an alternate method for investigating similar problems or checking solutions through a different lens.

Pythagorean Theorem

The Pythagorean theorem is one of the foundational results in geometry and it's highly likely that it's familiar even to beginners in mathematics. It states that in a right-angled triangle, the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.

This principle is crucial in solving the given problem, as it allows us to relate the sides of the right-angled triangles within the isosceles trapezoid to its diagonals. To apply this theorem effectively, one needs to identify right-angled triangles, label their sides appropriately, and apply the theorem to find relations between the lengths of these sides.

Geometric Congruence

Geometric congruence is a concept that tells us when two shapes have the same size and shape, allowing us to say that they are identical for all intents and purposes in geometry. Shapes that are congruent can be rotated, reflected, or translated, and they will still be considered congruent because their corresponding angles and side lengths are equal.

In the context of the given step-by-step solution, the congruence of two triangles within the trapezoid allows us to establish equalities between their respective sides and use those equalities to advance our proof. Understanding congruence is paramount for evaluating the properties of geometric figures and providing rigorous mathematical proofs.

Mathematical Proofs

Mathematical proofs are logical arguments that demonstrate the truth of a mathematical statement beyond any doubt. A proof is composed of a series of logical steps that proceed from a set of axioms, using agreed-upon mathematical principles, towards the statement that needs to be proved.

In proving that the square on the diagonal of an isosceles trapezoid equals the sum of the product of its parallel sides plus the square on one of the non-parallel sides, we use a direct proof. This involves applying the Pythagorean theorem to right triangles and using triangle congruence to link the lengths of the trapezoid's sides. Breaking down complex geometric problems into a series of clear and concrete steps is essential in mathematics, as it not only validates the solution but also aids in understanding.