Γ 1 is a circle with center O 1 and radius R 1 , Γ 2 is a circle with center O 2 and radius R 2 , and R 2 <R 1 . Γ 2 has O 1 on its circumference. O 1 O 2 intersect Γ 2 again at A . Circles Γ 1 and Γ 2 intersect at points B and C such that ∠CO 1 B=52 ∘ . D is a point on the circumference of Γ 1 that is not contained within Γ 2 . The line DB intersects Γ 2 at E . What is the measure (in degrees) of the acute angle between lines DE and EA ?
Posting Brilliant Questions!! You will not learn from it..such a spoiled brat!!
PLEASE SOLVE THE QUESTIONS..INSTEAD OF POSTING IT..BRILLIANT DOES NOT TOLERATE THIS KIND OF AKSING
To find the measure of the acute angle between lines DE and EA, we need to examine the given information and use some geometry concepts to solve the problem. Let's break down the problem step by step.
Step 1: Draw a diagram
Start by drawing a diagram based on the given information to visualize the scenario. You can draw two circles, Γ1 and Γ2, with center O1 and O2 respectively. Make sure O1 is on the circumference of Γ2. Label the radii R1 and R2, where R2 is smaller than R1. Draw the line O1O2 to connect the centers of the two circles. Label the point of intersection between O1O2 and Γ2 as A. Then, draw points B and C, which are the points of intersection between Γ1 and Γ2. Finally, draw point D on the circumference of Γ1 that is not inside Γ2, and draw the line DB, which intersects Γ2 at point E.
Step 2: Analyze the given angles
From the given information, we know that ∠CO1B = 52°. Locate point C on the diagram and label the angle ∠CO1B.
Step 3: Determine angle ∠CO1E
To find the measure of angle ∠CO1E, we need to identify the relationship between angles ∠CO1B and ∠CO1E. Notice that angle ∠CO1E is an inscribed angle of the circle Γ2, and angle ∠CO1B is an angle formed by a chord and the diameter of the same circle. By the inscribed angle theorem, we know that the measure of ∠CO1E is twice the measure of ∠CO1B. Therefore, ∠CO1E = 2 * 52° = 104°.
Step 4: Identify isosceles triangle AEO1
From the information given, we know that O1 is on the circumference of circle Γ2. By connecting points A, E, and O1, we form a triangle AEO1. It is an isosceles triangle because two sides, AO1 and EO1, have the same length (radius R2 of circle Γ2).
Step 5: Calculate angle ∠AEO1
Since triangle AEO1 is isosceles, the base angles ∠AEO1 and ∠EAO1 are equal. Using the fact that ∠CO1E = 104°, we can find ∠EAO1 as follows:
∠EAO1 = (180° - ∠CO1E)/2 = (180° - 104°)/2 = 76°.
Step 6: Find angle ∠DEA
To calculate the measure of angle ∠DEA, we need to consider that angle ∠DEA is an exterior angle of triangle AEO1. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. In this case, the remote interior angles are ∠EAO1 and ∠AEO1.
Thus, ∠DEA = ∠EAO1 + ∠AEO1 = 76° + 76° = 152°.
Therefore, the measure of the acute angle between lines DE and EA is 152°.