Two sides of a parallelogram measure 15 cm and 20 cm, and one of the diagonals measures 19 cm. W

Two sides of a parallelogram measure 15 cm and 20 cm, and...

Key Concepts

Trigonometric Inverse Functions

Trigonometric inverse functions, such as the inverse cosine, allow one to calculate an angle when the value of its cosine is known. This is crucial in situations where angles are deduced from side-length relationships, as found in applications of the Law of Cosines.

Properties of Diagonals in Parallelograms

In parallelograms, diagonals have the property of bisecting each other, and their lengths can be related to the side lengths and angles through geometric constructions. This relationship is key in analyzing and solving problems where diagonal measurements play a role in determining other elements of the figure.

Law of Cosines

The Law of Cosines is a mathematical tool that relates the lengths of sides in a triangle to the cosine of one angle. It is especially useful for solving triangles where traditional right-angle methods do not apply, and helps in finding unknown sides or angles given limited measurements.

Properties of Parallelograms

A parallelogram is a quadrilateral with both pairs of opposite sides being equal and parallel. This property implies that opposite angles are equal and consecutive angles are supplementary, which provides a framework for determining unknown angles when partial information is provided.

Transcript

00:01 So we'll start with our sketch of a parallelogram and we'll fill in information of length that we're given for this parallelogram.

00:11 We have the lengths of two of the sides and one of the diagonals.

00:18 So one side is 15 centimeters, the other side is 20 centimeters.

00:22 So we'll label the shorter side, 15 centimeters, and the longer side 20 centimeters.

00:30 Now we can draw another diagonal in here, separate from the blue diagonal that's already drawn, but we're actually given the length of 19 centimeters as the length of one of the diagonals.

00:53 The blue diagonal is going to be less than 20 centimeters.

01:00 And we can see based on the way the sketch is drawn.

01:04 And the red diagonal will be longer than 20 centimeters.

01:09 So the blue diagonal will label as 19 centimeters.

01:16 Now, if we view this parallelogram as two separate triangles, let's work with this triangle, the lower left triangle.

01:26 Since we have the length of the three sides of a triangle, we can use the law of cosines to calculate this angle here.

01:36 Of the parallelogram, the angle in the lower left.

01:40 And the law of cosines is c squared equals a squared plus b squared minus 2ab, cosine c.

01:50 When you go through and do some algebra on the formula for the law of cosines to solve for cosine c, you're left with that equal to a squared plus b squared minus c squared, all over 2ab.

02:12 And let's fill in the information...

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