You can use triangle congruence theorems to prove relationships among tangents and secants.
Task 1
Four tangents are drawn from E to two concentric circles. A, B, C, and D are the points of tangency. Name as many pairs of congruent triangles as possible and tell how you can show each pair is congruent.
Two concentric circles have center O, with point E outside. From O, segments lead to B and C on the inner circle and A and B on the outer circle, with segments AB and CD connecting the circles. Segments from E lead to A, B, O, C, and C.
BIG idea: Measurement
You can use facts about arcs and angle measures to solve real-world problems.
Task 2
The rocks near the shore between two lighthouses at points A and B make the waters unsafe. The measure of AXB⏜ modified eh x b with frown above is 300. Waters inside this arc are unsafe. Suppose you are a navigator on a ship at sea. How can you use the lighthouses to keep the ship in safe waters?
A portion of a circle connects points A and B on a shore, with point X on the portion closer to A.
Task 1:
To find pairs of congruent triangles in this scenario, we can use the fact that tangents to a circle from a point outside the circle are congruent. Let's go through each pair of triangles:
1. Triangle EAB and ECD: These triangles are congruent by the Side-Side-Side (SSS) congruence criterion. The sides EA, AB, and BE are congruent to the sides ED, CD, and DE, respectively, because they are tangent segments from the same point E.
2. Triangle ECB and EBA: These triangles are also congruent by the SSS congruence criterion. The sides EC, CB, and BE are congruent to the sides EA, AB, and BE, respectively, because they are tangent segments from the same point E.
3. Triangle EOA and EOC: These triangles are congruent by the Side-Angle-Side (SAS) congruence criterion. The sides EO and EO are congruent because they are radii of the circle, and the included angles OAE and OCE are congruent because they are both right angles.
Therefore, we have identified three pairs of congruent triangles: EAB and ECD, ECB and EBA, and EOA and EOC.
Task 2:
To keep the ship in safe waters using the lighthouses, we need to ensure that the ship stays outside the unsafe arc AXB. Here's how we can do that:
1. Determine the bearing of each lighthouse: Lighthouses at points A and B can be used as reference points. As a navigator, you can determine the bearing or direction of each lighthouse relative to your ship's current position.
2. Keep the ship between the bearings: By knowing the direction or bearing of each lighthouse, you can steer the ship to keep it within a safe zone between the bearings. The ship should stay on the side of the unsafe arc opposite to the lighthouse it is nearest to.
For example, if the bearing of lighthouse A is 10 degrees and the bearing of lighthouse B is 50 degrees, the ship should maintain a heading between 10 degrees and 50 degrees to stay clear of the unsafe waters in the arc AXB.