The Converse of the Pythagorean Theorem

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Presentation transcript:

The Converse of the Pythagorean Theorem

What is the converse of a theorem? If a theorem is pq (if p then q) The converse of that theorem is qp (if q then p)

A triangle has sides a,b, and c such that a2+b2=c2 IF A triangle has sides a,b, and c such that a2+b2=c2 THEN So what is the converse of the Pythagorean Theorem? It is a right triangle

Practice 5 m 4 m 11 cm 6 cm 5 cm 8 cm These figures are not drawn accurately. Which of the triangles are right triangles

8 cm 6 cm 10 cm 6 cm 4 cm 5 cm You Do: 2 cm 7 cm 12 cm These figures are not drawn accurately, which of these triangles are right?

Applications Pythagorean Theorem

Steps for application problems: Draw a clear diagram of the situation Label your diagram with information you were given Determine which measurement you need to find. Label this “x” Use what you are given to find x.

A ladder leans against a house. The feet of the ladder are 2 A ladder leans against a house. The feet of the ladder are 2.1 m from the base of the wall. The top of the ladder hits the house 3.8 m above the ground. How long is the ladder? 3.8 m x m 2.1 m Practice Together

You Do: A rectangular gate is 3 m wide and has a 3.5 m diagonal. How high is the gate? 3 m 3.5 m x m

A surveyor makes the following measurements of a field A surveyor makes the following measurements of a field. Calculate the perimeter of the field to the nearest metre. 8 m 11 m 3 m 11 m 17 m Practice Together

You Do: Satomi says she has just cut out a triangular sail for her boat. The lengths of the sides are 6.23 m, 3.87 m and 4.88 m. The sail is supposed to be right angled. Is it?

Circle Problems: Chord of a Circle The line drawn from the centre of a circle at right angles to a chord bisects the chord. chord radius centre

Circle Problem: Chord of a Circle Practice Together Circle Problem: Chord of a Circle A chord of length 8 cm is 4 cm from the centre of a circle. Find the length of the circle’s radius

Circle Problem: Chord of a Circle You Do: Circle Problem: Chord of a Circle A chord of a circle has length 2 cm and the circle has radius 3 cm. Find the shortest distance from the centre of the circle to the chord.

Circle Problems: Tangent-Radius Property A tangent to a circle and a radius at the point of contact meet at right angles radius centre point of contact tangent

Circle Problem: Tangent-Radius Property If the earth has a radius of 6400 km and you are in a rocket 50 km directly above the earth’s surface, determine the distance to the horizon. 50 km Practice Together: Distance in the horizon 6400 km 6400 km

Circle Problem: Tangent-Radius Property A circle has radius 2 cm. A tangent is drawn to the circle from point P, which is 9 cm from O, the circle’s centre. How long is the tangent? O 9 cm 2 cm You Do: P

Three-Dimensional Problem A room is 6 m by 5 m and has a height of 3 m. Find the distance from a corner point on the floor to the opposite corner of the building 3 m 5 m 6 m Practice Together:

You Do: Three-Dimensional Problems Choose a cylindrical container and find the length of the longest nail that could be put entirely within the container. You Do: