similarities in a right triangle
DESCRIPTION
SIMILARITIES IN A RIGHT TRIANGLE. By: SAMUEL M. GIER. How much do you know. DRILL. SIMPLIFY THE FOLLOWING EXPRESSION. 1. 4. + 2. 5 . 3. DRILL. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: SIMILARITIES IN A RIGHT TRIANGLE](https://reader035.vdocuments.mx/reader035/viewer/2022070404/56813adf550346895da32b5b/html5/thumbnails/1.jpg)
SIMILARITIES IN A RIGHT TRIANGLE
By: SAMUEL M. GIER
![Page 2: SIMILARITIES IN A RIGHT TRIANGLE](https://reader035.vdocuments.mx/reader035/viewer/2022070404/56813adf550346895da32b5b/html5/thumbnails/2.jpg)
How much do you know
![Page 3: SIMILARITIES IN A RIGHT TRIANGLE](https://reader035.vdocuments.mx/reader035/viewer/2022070404/56813adf550346895da32b5b/html5/thumbnails/3.jpg)
DRILL
SIMPLIFY THE FOLLOWING EXPRESSION.
1. 4.
+
2.
5.
3.
42
45
72
64 4
464
![Page 4: SIMILARITIES IN A RIGHT TRIANGLE](https://reader035.vdocuments.mx/reader035/viewer/2022070404/56813adf550346895da32b5b/html5/thumbnails/4.jpg)
DRILL
Find the geometric mean between the two given numbers.
1. 6 and 8
2. 9 and 4
![Page 5: SIMILARITIES IN A RIGHT TRIANGLE](https://reader035.vdocuments.mx/reader035/viewer/2022070404/56813adf550346895da32b5b/html5/thumbnails/5.jpg)
DRILL
Find the geometric mean between the two given numbers.
1. 6 and 8
h=
=
=
h= 4
)8(6
48
)3(16
3
![Page 6: SIMILARITIES IN A RIGHT TRIANGLE](https://reader035.vdocuments.mx/reader035/viewer/2022070404/56813adf550346895da32b5b/html5/thumbnails/6.jpg)
DRILL
Find the geometric mean between the two given numbers.
2. 9 and 4
h=
=
h= 6
)4(9
36
![Page 7: SIMILARITIES IN A RIGHT TRIANGLE](https://reader035.vdocuments.mx/reader035/viewer/2022070404/56813adf550346895da32b5b/html5/thumbnails/7.jpg)
REVIEW ABOUT RIGHT TRIANGLES
ABBC
A
CB
LEGS
AC
&
HYPOTENUSE
The side opposite the right angle
The perpendicular side
![Page 8: SIMILARITIES IN A RIGHT TRIANGLE](https://reader035.vdocuments.mx/reader035/viewer/2022070404/56813adf550346895da32b5b/html5/thumbnails/8.jpg)
SIMILARITIES IN A RIGHT TRIANGLE
By: SAMUEL M. GIER
![Page 9: SIMILARITIES IN A RIGHT TRIANGLE](https://reader035.vdocuments.mx/reader035/viewer/2022070404/56813adf550346895da32b5b/html5/thumbnails/9.jpg)
CONSIDER THIS…
State the means and the extremes in the following statement.
3:7 = 6:14
The means are 7 and 6 and the extremes are 3 and 14.
![Page 10: SIMILARITIES IN A RIGHT TRIANGLE](https://reader035.vdocuments.mx/reader035/viewer/2022070404/56813adf550346895da32b5b/html5/thumbnails/10.jpg)
CONSIDER THIS…
State the means and the extremes in the following statement.
5:3 = 6:10
The means are 3 and 6 and the extremes are 5 and 10.
![Page 11: SIMILARITIES IN A RIGHT TRIANGLE](https://reader035.vdocuments.mx/reader035/viewer/2022070404/56813adf550346895da32b5b/html5/thumbnails/11.jpg)
CONSIDER THIS…
State the means and the extremes in the following statement.
a:h = h:b
The means are h and the extremes are a and b.
![Page 12: SIMILARITIES IN A RIGHT TRIANGLE](https://reader035.vdocuments.mx/reader035/viewer/2022070404/56813adf550346895da32b5b/html5/thumbnails/12.jpg)
CONSIDER THIS…
Find h.
a:h = h:b
applying the law of proportion. h² = ab
h= ab
h is the geometric mean between a & b.
![Page 13: SIMILARITIES IN A RIGHT TRIANGLE](https://reader035.vdocuments.mx/reader035/viewer/2022070404/56813adf550346895da32b5b/html5/thumbnails/13.jpg)
THEOREM:SIMILARITIES IN A RIGHT
TRIANGLE
States that “In a right triangle, the altitude to the hypotenuse separates the triangle into two triangles each similar to the given triangle and to each other.
![Page 14: SIMILARITIES IN A RIGHT TRIANGLE](https://reader035.vdocuments.mx/reader035/viewer/2022070404/56813adf550346895da32b5b/html5/thumbnails/14.jpg)
ILLUSTRATION
“In a right triangle (∆MOR), the altitude to the hypotenuse(OS) separates the triangle into two triangles(∆MOS & ∆SOR )each similar to the given triangle (∆MOR) and to each other.
M
S
RO
∆MOR ~ ∆MSO, ∆MOR ~ ∆OSR by AA Similarity postulate)
∆MSO~ ∆OSR by transitivity
![Page 15: SIMILARITIES IN A RIGHT TRIANGLE](https://reader035.vdocuments.mx/reader035/viewer/2022070404/56813adf550346895da32b5b/html5/thumbnails/15.jpg)
TRY THIS OUT!
NAME ALL SIMILAR TRIANGLES
A
B
DC
∆ACD ~ ∆ABC∆ACD ~ ∆CBD∆ABC ~ ∆CBD
![Page 16: SIMILARITIES IN A RIGHT TRIANGLE](https://reader035.vdocuments.mx/reader035/viewer/2022070404/56813adf550346895da32b5b/html5/thumbnails/16.jpg)
COROLLARY 1.
In a right triangle, the altitude to the hypotenuse is the geometric
mean of the segments into which it divides the hypotenuse
![Page 17: SIMILARITIES IN A RIGHT TRIANGLE](https://reader035.vdocuments.mx/reader035/viewer/2022070404/56813adf550346895da32b5b/html5/thumbnails/17.jpg)
ILLUSTRATION
CB is the geometric mean between AB & BD.
A
B
DC
In the figure,
BD
CB
CB
AB
![Page 18: SIMILARITIES IN A RIGHT TRIANGLE](https://reader035.vdocuments.mx/reader035/viewer/2022070404/56813adf550346895da32b5b/html5/thumbnails/18.jpg)
COROLLARY 2.
In a right triangle, either leg is the geometric mean between the hypotenuse and the segment of the hypotenuse adjacent to it.
![Page 19: SIMILARITIES IN A RIGHT TRIANGLE](https://reader035.vdocuments.mx/reader035/viewer/2022070404/56813adf550346895da32b5b/html5/thumbnails/19.jpg)
ILLUSTRATION
CB is the geometric mean between AB & BD.
A
B
DC
In the figure,
AD
CD
CD
DB
AD
CA
CA
AB