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No cheating!
The triangle is a right angled triangle with sides measuring 3 (and 4) and the hypotenuse of length 5. Note - The length of the third side bc = 4, can be calculated using Pythagoras Theorem. If d is 1 unit of length along the hypotenuse and a perpendicular line is drawn from bc to d (meeting bc at e) then the triangle bde is similar to triangle bac. Then ca/ba = 3/5 = ed/bd = ed/1. Thus ed = 3/5 units in length. The area of a triangle = ½ x base x vertical height. The area of triangle dbc = ½ x bc x ed = ½ x 4 x 3/5 = 1.2 sq units.
Triangles BDC, CBD, CDB, DBC and DCB.
To prove that triangle ABC is congruent to triangle DBC given that line CB bisects angle ACD, we can use the Angle-Side-Angle (ASA) postulate. Proof: | Statements | Reasons | |---------------------------------------------------|------------------------------------------| | 1. CB bisects angle ACD. | 1. Given. | | 2. Angle ACB ≅ Angle DCB. | 2. Definition of angle bisector. | | 3. Line CB is common to both triangles ABC and DBC. | 3. Common side. | | 4. Triangle ABC ≅ Triangle DBC. | 4. ASA Postulate (Angle ACB ≅ Angle DCB, CB is common, and Angle ACD is shared). |
If triangle ABC is similar to triangle DBC, and segment BC bisects angle ACD, it indicates that triangles ACB and DBC share proportional relationships. This means that the angles and sides of the triangles are related in a consistent manner. The bisector implies that the angles created by the bisector are equal, further establishing the similarity and proportionality between the two triangles. Thus, the properties of angle bisectors and similar triangles can be applied to analyze the geometric relationships within this configuration.
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Angle ABD = 4x - 4 Angle ABC = twice angle ABD = 7x + 4 So 7x + 4 = 2*(4x - 4) = 8x - 8 So x = 12 Then angle DBC = half of angle ABC = 1/2*(7*12 + 4) = 1/2*88 = 44 degrees.
The measurement of angle ABD is 73 degrees. You find this angle by subtracting angle DBC from angle ABC, or 89-16 is equal to 73 degrees.
In a circle, the angle subtended by a diameter at the circumference is a right angle. Given that angle BAC is 50 degrees, and angle BAD is 105 degrees, we can find angle DBC by first finding angle ABC. Since angle ABC is the external angle to triangle ABD, it can be calculated as BAD - BAC = 105 - 50 = 55 degrees. Therefore, angle DBC, which is equal to angle ABC, is 55 degrees.
The answer is 13. x=13 13*5+13*2=91 Thank you.
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No cheating!
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DBC Pierre was born in 1961.
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