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Foundations of Algebra :

SIMPLIFYING RADICALS

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Simplify $\mathbf{\color{green}{ \sqrt{24\,}\,\sqrt{6\,} }}$

Neither of 24 and 6 is a square, but what happens if I multiply them inside one radical?

$\sqrt{24\,}\,\sqrt{6\,} = \sqrt{24\times 6\,} = \sqrt{144\,}$

Now I *do* have something with squares in it, so I can simplify as before:

$\sqrt{144\,} = \sqrt{12 \times 12\,} = \mathbf{\color{purple}{ 12 }}$

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Simplify $\mathbf{\color{green}{ \sqrt{75\,} }}$

The argument of this radical, 75, factors as:

This factorization gives me two copies of the factor 5, but only one copy of the factor 3. Since I have two copies of 5, I can take 5 out front. Since I have only the one copy of 3, it'll have to stay behind in the radical. Then my answer is:

$\sqrt{75\,} = \sqrt{3\times 25}$

$= \sqrt{3\,}\,(5) = \mathbf{\color{purple}{ 5\sqrt{3\,} }}$

This answer is pronounced as "five, times root three", "five, times the square root of three", or, most commonly, just "five, root three".

Anatomy

CONNECTIVE TISSSUES

§Connective tissues have diverse
structures and functions (continued)

–Specialized connective tissues–This diverse group includes cartilage,
bone, fat, blood, and lymph–Cartilage
consists of widely spaced cells surrounded by a thick, nonliving matrix
composed of collagen secreted by the cartilage cells–Bone
resembles cartilage, but its matrix is hardened by deposits of calcium
phosphate–Adipose tissue is
made up of fat cells that are modified for long-term energy storage

–Adipose tissue can also serve as
insulation for animals living in a cold environment

Geometry

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How to tell if triangles are congruent

Any triangle is defined by six measures (three sides, three angles). But you don't need to know all of them to show that two triangles are congruent. Various groups of three will do. Triangles are congruent if:
**SSS** (side side side)

All three corresponding sides are equal in length.

See Triangle Congruence (side side side).
**SAS **(side angle side)

A pair of corresponding sides and the included angle are equal.

See Triangle Congruence (side angle side).
**ASA **(angle side angle)

A pair of corresponding angles and the included side are equal.

See Triangle Congruence (angle side angle).
**AAS **(angle angle side)

A pair of corresponding angles and a non-included side are equal.

See Triangle Congruence (angle angle side).
**HL **(hypotenuse leg of a right triangle)

Two right triangles are congruent if the hypotenuse and one leg are equal.

See Triangle Congruence (hypotenuse leg).

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AAA does not work.

If all the corresponding angles of a triangle are the same, the triangles will be the same shape, but not necessarily the same size. For more on this see

Why AAA doesn't work.

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SSA does not work.

Given two sides and a non-included angle, it is possible to draw two different triangles that satisfy the the values. It is therefore not sufficient to prove congruence. See

Why SSA doesn't work.

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Constructions

Another way to think about the above is to ask if it is possible to construct a unique triangle given what you know. For example, If you were given the lengths of two sides and the included angle (SAS), there is only one possible triangle you could draw. If you drew two of them, they would be the same shape and size - the definition of congruent. For more on constructions, see Introduction to Constructions##
Properties of Congruent Triangles

If two triangles are congruent, then each part of the triangle (side or angle) is congruent to the corresponding part in the other triangle. This is the true value of the concept; once you have proved two triangles are congruent, you can find the angles or sides of one of them from the other.

To remember this important idea, some find it helpful to use the acronym

CPCTC, which stands for "

**C**orresponding

**P**arts of

**C**ongruent

**T**riangles are

**C**ongruent".

In addition to sides and angles, all other properties of the triangle are the same also, such as area, perimeter, location of centers, circles etc.