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In right triangle ABC , where angle c is the right angle , point D lies on sides AB, and point E lies on side BC such that AC=CD, DE=EB, and the ratio AC :DE = 4:3 .what is the ratio of AD:BD?
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In right triangle ABC , where angle c is the right angle , point D lie...
Given Information:
- Right triangle ABC with angle C as the right angle.
- Point D lies on side AB and point E lies on side BC.
- AC = CD, DE = EB, and AC : DE = 4 : 3.
Ratio of AD : BD:
To find the ratio of AD : BD, we need to consider the given information and apply the properties of similar triangles.
Property 1:
Since AC = CD, triangle ACD is an isosceles triangle. Therefore, angle ACD = angle ADC.
Property 2:
Since DE = EB, triangle DEB is an isosceles triangle. Therefore, angle DEB = angle EDB.
Property 3:
From the given ratio AC : DE = 4 : 3, we can write AC = 4x and DE = 3x for some constant x.
Property 4:
In triangle ACD and triangle AEB, we have:
- angle ACD = angle AEB (both are right angles)
- angle ADC = angle ADE (from Property 1)
- angle ACD = angle AEB (from Property 2)
Property 5:
Using the property of similar triangles, we can write:
AC/AD = DE/DB = CD/CE
Calculation of Ratio AD : BD:
From Property 3, we have AC = 4x and DE = 3x. Therefore, CD = AC - AD = 4x - AD and CE = DE + EB = 3x + EB.
Substitute these values into the ratios from Property 5:
4x/AD = 3x/(3x + EB) = (4x - AD)/(3x + EB)
Solving these equations will give you the ratio of AD : BD.
- Right triangle ABC with angle C as the right angle.
- Point D lies on side AB and point E lies on side BC.
- AC = CD, DE = EB, and AC : DE = 4 : 3.
Ratio of AD : BD:
To find the ratio of AD : BD, we need to consider the given information and apply the properties of similar triangles.
Property 1:
Since AC = CD, triangle ACD is an isosceles triangle. Therefore, angle ACD = angle ADC.
Property 2:
Since DE = EB, triangle DEB is an isosceles triangle. Therefore, angle DEB = angle EDB.
Property 3:
From the given ratio AC : DE = 4 : 3, we can write AC = 4x and DE = 3x for some constant x.
Property 4:
In triangle ACD and triangle AEB, we have:
- angle ACD = angle AEB (both are right angles)
- angle ADC = angle ADE (from Property 1)
- angle ACD = angle AEB (from Property 2)
Property 5:
Using the property of similar triangles, we can write:
AC/AD = DE/DB = CD/CE
Calculation of Ratio AD : BD:
From Property 3, we have AC = 4x and DE = 3x. Therefore, CD = AC - AD = 4x - AD and CE = DE + EB = 3x + EB.
Substitute these values into the ratios from Property 5:
4x/AD = 3x/(3x + EB) = (4x - AD)/(3x + EB)
Solving these equations will give you the ratio of AD : BD.
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In right triangle ABC , where angle c is the right angle , point D lies on sides AB, and point E lies on side BC such that AC=CD, DE=EB, and the ratio AC :DE = 4:3 .what is the ratio of AD:BD?
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