hectictar

This diagram has more information than I showed how to find. If you're unsure how to find, for example,  AD = 30, and want further explanation, just ask.

m∠EDC  =  m∠ABC   because corresponding angles are congruent.

m∠ECD  =  m∠ACB   because they are the same angle

So by the AA similarity theorem,  △EDC ~ △ABC

When we draw a height to the base of an isosceles triangle, we split the base exactly in half.

So the scale factor from △ABC  to  △EDC  is  1/2.  And so...

area of △EDC  =  ( 1/2 )( 1/2 )( area of △ABC)

area of ABDE   =   area of △ABC - area of △EDC

area of ABDE   =   180 - ( 1/2 )( 1/2 )( 180 )

area of ABDE   =   180 - 45

area of ABDE   =   135

The areas are in square units.

Yep! That's what I got too

If the dot is raised like this  ·  it is a multiplication symbol.

If the dot is on the floor like this  .  it is a decimal point.

7·8  means  "7 times 8"

7.8  means  "7 point 8"  which is  7  and  8 tenths

Let's just look at the similarity statement  △ABD ~ △CBA

△ABD ~ △CBA     AB  in the first triangle corresponds to  CB  in the second triangle.

△ABD ~ △CBA     BD  in the first triangle corresponds to  BA  in the second triangle.

So:

AB / CB  =  BD / BA

Do you see where that came from?

All that's left to do is plug in the lengths of those sides and solve for  y .

m∠ABD  =  m∠CBA  because they are the same angle

m∠BDA  =  m∠BAC  because they are both right angles.

So by the AA similarity theorem, triangle ABD is similar to triangle CBA

Do you see what to do next?

m∠ABD + m∠CBD  =  90°

m∠ABD + m∠BAD  =  90°

So  m∠CBD =  m∠BAD

And  m∠BDA  =  m∠CDB  because they are both right angles.

So by the AA similarity theorem,  △ABD  ~ △BCD

BD / CD  =  AD / BD

x / 10  =  4 / x

x² / 10  =  4

x²  =  40

x  =  sqrt( 40 )

x  =  2 sqrt( 10 )

Let     f(x)  =  x * (3x + 1)  =  3x² + x

Let's see what  f(x)   is  when  x  is at the endpoints of the interval.

f(-7/3)  =  3(-7/3)² + (-7/3)  =  14

f(2)  =  3(2)² + 2  =  14

Aha! they are the same, just as I suspected! 🕵️‍♀️

Let's see what  f(x)  is when  x  is in the interval.

f(0)  =  3(0)² + 0  =  0

And it is true that  0 < 14

Since  f(x)  is a parabola, we can be sure that  f(x) < 14  if and only if  x  is in the interval  (-7/3, 2)

Remember, the radius of a circle is the distance between the center and a point on the circle.

The center of the circle is  (2, 3)

The point  (-2, 0)  is on the circle.

Let's use the distance formula to find the radius.

radius  =  distance between  (2, 3)  and  (-2, 0)

radius  =  \(\sqrt{(0-3)^2+(-2-2)^2}\)

radius  =  \(\sqrt{(-3)^2+(-4)^2}\)

radius  =  \(\sqrt{9+16}\)

radius  =  \(\sqrt{25}\)

radius  =  5